156 results on '"49Q15"'
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2. A modified Fletcher-Reeves conjugate gradient method for unconstrained optimization with applications in image restoration.
- Author
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Ahmed, Zainab Hassan, Hbaib, Mohamed, and Abbo, Khalil K.
- Subjects
- *
CONJUGATE gradient methods , *ALGORITHMS - Abstract
The Fletcher-Reeves (FR) method is widely recognized for its drawbacks, such as generating unfavorable directions and taking small steps, which can lead to subsequent poor directions and steps. To address this issue, we propose a modification to the FR method, and then we develop it into the three-term conjugate gradient method in this paper. The suggested methods, named "HZF" and "THZF", preserve the descent property of the FR method while mitigating the drawbacks. The algorithms incorporate strong Wolfe line search conditions to ensure effective convergence. Through numerical comparisons with other conjugate gradient algorithms, our modified approach demonstrates superior performance. The results highlight the improved efficacy of the HZF algorithm compared to the FR and three-term FR conjugate gradient methods. The new algorithm was applied to the problem of image restoration and proved to be highly effective in image restoration compared to other algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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3. Partial Plateau's Problem with $H$-mass
- Author
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Alvarado, Enrique and Xia, Qinglan
- Subjects
Mathematics - Classical Analysis and ODEs ,Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,49Q15 - Abstract
Classically, Plateau's problem asks to find a surface of the least area with a given boundary $B$. In this article, we investigate a version of Plateau's problem, where the boundary of an admissible surface is only required to partially span $B$. Our boundary data is given by a flat $(m-1)$-chain $B$ and a smooth compactly supported differential $(m-1)$-form $\Phi$. We are interested in minimizing $ \mathbf{M}(T) - \int_{\partial T} \Phi $ over all $m$-dimensional rectifiable currents $T$ in $\mathbb{R}^n$ such that $\partial T$ is a subcurrent of the given boundary $B$. The existence of a rectifiable minimizer is proven with Federer and Fleming's compactness theorem. We generalize this problem by replacing the mass $\mathbf{M}$ with the $H$-mass of rectifiable currents. By minimizing over a larger class of objects, called scans with boundary, and by defining their $H$-mass as a type of lower-semicontinuous envelope over the $H$-mass of rectifiable currents, we prove an existence result for this problem by using Hardt and De Pauw's BV compactness theorem., Comment: 22 pages, 5 figures
- Published
- 2023
4. Integral decompositions of varifolds
- Author
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Chou, Hsin-Chuang
- Subjects
Mathematics - Differential Geometry ,49Q15 - Abstract
This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is established. Furthermore, this result can be generalized by replacing the class of integral varifolds by some classes of rectifiable varifolds whose density is uniformly bounded from below. However, such decomposition may fail to be unique., Comment: 15 pages
- Published
- 2022
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5. The regularity theory for the area functional (in geometric measure theory)
- Author
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De Lellis, Camillo
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,49Q15 - Abstract
The aim of this article is to give a rather extensive, and yet nontechnical, account of the birth of the regularity theory for generalized minimal surfaces, of its various ramifications along the decades, of the most recent developments, and of some of the remaining challenges., Comment: To appear in the Proceedings of the ICM
- Published
- 2021
6. On the lack of compactness in the axisymmetric neo-Hookean model
- Author
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Marco Barchiesi, Duvan Henao, Carlos Mora-Corral, and Rémy Rodiac
- Subjects
49J45 ,49Q15 ,74B20 ,74G65 ,74G70 ,Mathematics ,QA1-939 - Abstract
We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti and De Lellis is generic in some sense. On this map, we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of $\mathbb {S}^2$ -valued harmonic maps.
- Published
- 2024
- Full Text
- View/download PDF
7. Second Variation of Sub-Riemannian Surface Measure of Non-horizontal Submanifolds in Sub-Riemannian Stratified Lie Groups.
- Author
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Santos, Maria R. B. and Veloso, José M. M.
- Subjects
- *
LIE groups , *MINIMAL surfaces , *SUBMANIFOLDS , *PARABOLOID - Abstract
We establish the second variation of sub-Riemannian surface measure for minimal non-horizontal submanifolds of a sub-Riemannian stratified Lie group. We obtain some applications for codimension one. Furthermore, we present a new proof of the fact that the hyperbolic paraboloid is stable in the Heisenberg group. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
8. On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions
- Author
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Bellettini, Giovanni, Elshorbagy, Alaa, Paolini, Maurizio, and Scala, Riccardo
- Subjects
Mathematics - Analysis of PDEs ,49Q15 - Abstract
In this paper we estimate from above the area of the graph of a singular map $u$ taking a disk to three vectors, the vertices of a triangle, and jumping along three $\mathcal{C}^2-$ embedded curves that meet transversely at only one point of the disk. We show that the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to "fill the hole" in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of $u$, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of $u$ cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections., Comment: 35 pages and 15 figures
- Published
- 2019
9. Total mean curvatures of Riemannian hypersurfaces
- Author
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Ghomi Mohammad and Spruck Joel
- Subjects
reilly’s formulas ,quermassintegral ,mixed volume ,generalized mean curvature ,hyperbolic space ,cartan-hadamard manifold ,primary: 53c20 ,58j05 ,secondary: 52a38 ,49q15 ,Mathematics ,QA1-939 - Abstract
We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ\Gamma in a Cartan-Hadamard manifold MM. In particular, we show that the first mean curvature integral of a convex hypersurface γ\gamma nested inside Γ\Gamma cannot exceed that of Γ\Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ\Gamma in terms of the volume it bounds in MM in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ\gamma is parallel to Γ\Gamma , or MM has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.
- Published
- 2023
- Full Text
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10. Lipschitz Chain Approximation of Metric Integral Currents
- Author
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Goldhirsch Tommaso
- Subjects
metric spaces ,integral currents ,polyhedral chains ,lipschitz chains ,49q15 ,58a25 ,53c23 ,Analysis ,QA299.6-433 - Abstract
Every integral current in a locally compact metric space X can be approximated by a Lipschitz chain with respect to the normal mass, provided that Lipschitz maps into X can be extended slightly.
- Published
- 2022
- Full Text
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11. Multiple valued sections of vector bundles: the reparametrization theorem for $Q$-valued functions revisited
- Author
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Stuvard, Salvatore
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,49Q15 - Abstract
We analyze a notion of multiple valued sections of a vector bundle over an abstract smooth Riemannian manifold, which was suggested by W. Allard in the unpublished note "Some useful techniques for dealing with multiple valued functions" and generalizes Almgren's $Q$-valued functions. We study some relevant properties of such $Q$-multisections and apply the theory to provide an elementary and purely geometric proof of a delicate reparametrization theorem for multi-valued graphs which plays an important role in the regularity theory for higher codimension area minimizing currents \`a la Almgren-De Lellis-Spadaro., Comment: V2 is the final version, to appear in Comm. Anal. Geom
- Published
- 2017
- Full Text
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12. Existence results for minimizers of parametric elliptic functionals
- Author
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De Philippis, Guido, De Rosa, Antonio, and Ghiraldin, Francesco
- Subjects
Mathematics - Analysis of PDEs ,49Q15 - Abstract
We prove a compactness principle for the anisotropic formulation of the Plateau problem in any codimension, in the same spirit of the previous works of the authors \cite{DelGhiMag,DePDeRGhi,DeLDeRGhi16}. In particular, we perform a new strategy for the proof of the rectifiability of the minimal set, based on the new anisotropic counterpart of the Allard rectifiability theorem proved by the authors in \cite{DePDeRGhi2}. As a consequence we provide a new proof of Reifenberg existence theorem.
- Published
- 2017
13. A Geometric Integration Approach to Nonsmooth, Nonconvex Optimisation.
- Author
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Riis, Erlend S., Ehrhardt, Matthias J., Quispel, G. R. W., and Schönlieb, Carola-Bibiane
- Subjects
- *
GEOMETRIC approach , *SUPERVISED learning , *NONSMOOTH optimization , *ENERGY dissipation , *PROBLEM solving , *BILEVEL programming , *NUMERICAL integration - Abstract
The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We present efficient and robust derivative-free methods called randomised Itoh–Abe methods. These are generalisations of the Itoh–Abe discrete gradient method, a well-known scheme from geometric integration, which has previously only been considered in the smooth setting. We demonstrate that the method and its favourable energy dissipation properties are well defined in the nonsmooth setting. Furthermore, we prove that whenever the objective function is locally Lipschitz continuous, the iterates almost surely converge to a connected set of Clarke stationary points. We present an implementation of the methods, and apply it to various test problems. The numerical results indicate that the randomised Itoh–Abe methods can be superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while still remaining competitive in terms of efficiency. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
14. On the structure of flat chains modulo $p$
- Author
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Marchese, Andrea and Stuvard, Salvatore
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Functional Analysis ,49Q15 - Abstract
In this paper, we prove that every equivalence class in the quotient group of integral $1$-currents modulo $p$ in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for $m$-dimensional integral currents modulo $p$ implies that the family of $(m-1)$-dimensional flat chains of the form $pT$, with $T$ a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for $0$-dimensional flat chains, and, using a proposition from "The structure of minimizing hypersurfaces mod $4$" by Brian White, also for flat chains of codimension $1$., Comment: 19 pages. Final version, to appear in Adv. Calc. Var
- Published
- 2016
- Full Text
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15. Semigenerated Carnot algebras and applications to sub-Riemannian perimeter.
- Author
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Le Donne, Enrico and Moisala, Terhi
- Abstract
This paper contributes to the study of sets of finite intrinsic perimeter in Carnot groups. Our intent is to characterize in which groups the only sets with constant intrinsic normal are the vertical half-spaces. Our viewpoint is algebraic: such a phenomenon happens if and only if the semigroup generated by each horizontal half-space is a vertical half-space. We call semigenerated those Carnot groups with this property. For Carnot groups of nilpotency step 3 we provide a complete characterization of semigeneration in terms of whether such groups do not have any Engel-type quotients. Engel-type groups, which are introduced here, are the minimal (in terms of quotients) counterexamples. In addition, we give some sufficient criteria for semigeneration of Carnot groups of arbitrary step. For doing this, we define a new class of Carnot groups, which we call type (◊) and which generalizes the previous notion of type (⋆) defined by M. Marchi. As an application, we get that in type (◊) groups and in step 3 groups that do not have any Engel-type algebra as a quotient, one achieves a strong rectifiability result for sets of finite perimeter in the sense of Franchi, Serapioni, and Serra-Cassano. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
16. Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces
- Author
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Pankka Pekka and Soultanis Elefterios
- Subjects
metric currents ,bld-mappings ,30l10 ,49q15 ,30c65 ,Analysis ,QA299.6-433 - Abstract
Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f* : Mk,loc(Y) → Mk,loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f* : Mk,loc(X) → Mk,loc(Y).
- Published
- 2019
- Full Text
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17. The pointed flat compactness theorem for locally integral currents
- Author
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Lang, Urs and Wenger, Stefan
- Subjects
Mathematics - Differential Geometry ,Mathematics - Metric Geometry ,49Q15 ,53C23 - Abstract
Recently, a new embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric spaces. To this end we introduce another variant of the Ambrosio--Kirchheim theory of currents in metric spaces, including currents with finite mass in bounded sets., Comment: 24 pages
- Published
- 2010
18. Almost complex structures and calibrated integral cycles in contact 5-manifolds
- Author
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Bellettini, Costante
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Symplectic Geometry ,49Q05 ,49Q15 ,53D10 ,32Q65 - Abstract
In a contact manifold (M^5, alpha), we consider almost complex structures J which satisfy, for any vector v in the horizontal distribution, d alpha (v,Jv) = 0. We prove that integral cycles whose approximate tangent planes have the property of being J-invariant are in fact smooth Legendrian curves except possibly at isolated points and we investigate how such structures J are related to calibrations., Comment: 35 pages
- Published
- 2009
19. Isoperimetric, Sobolev and Poincar\'e inequalities on hypersurfaces in sub-Riemannian Carnot groups
- Author
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Montefalcone, F.
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,49Q15 ,46E35 ,22E60 - Abstract
In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or without - boundary $\partial S$; S and $\partial S$ are endowed with their homogeneous measures, actually equivalent to the intrinsic (Q-1)-dimensional and (Q-2)-dimensional Hausdorff measures with respect to some homogeneous metric $\varrho$ on G; see Section 5. This generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [63] and, independently by Allard [1]. In particular, from this result one may deduce some related Sobolev-type inequalities; see Section 7. The strategy of the proof is inspired by the classical one. In particular, we shall begin by proving some linear isoperimetric inequalities. Once this is proven, one can deduce a local monotonicity formula and then conclude the proof by a covering argument. We stress however that there are many differences, due to our different geometric setting. Some of the tools which have been developed ad hoc in this paper are, in order, a ``blow-up'' theorem, which also holds for characteristic points, and a smooth Coarea Formula for the HS-gradient; see Section 3 and Section 4. Other tools are the horizontal integration by parts formula and the 1st variation of the H-perimeter already developed in [68], [69], and here generalized to hypersurfaces having non-empty characteristic set. Some natural applications of these results are in the study of minimal and constant horizontal mean curvature hypersurfaces. Moreover we shall prove some purely horizontal, local and global Poincar\'e-type inequalities as well as some related facts and consequences; see Section 4 and Section 5., Comment: 61 pages
- Published
- 2009
20. CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian Carnot groups
- Author
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Arcozzi, N., Ferrari, F., and Montefalcone, F.
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,49Q15 ,46E35 ,22E60 - Abstract
In this paper we study the main geometric properties of the Carnot-Carath\'eodory (abbreviated CC) distance $\dc$ in the setting of $k$-step sub-Riemannian Carnot groups from many different points of view. An extensive study of the so-called normal CC-geodesics is given. We state and prove some related variational formulae and we find suitable Jacobi-type equations for normal CC-geodesics. One of our main results is a sub-Riemannian version of the Gauss Lemma. We show the existence of the metric normal for smooth non-characteristic hypersurfaces. We also compute the sub-Riemannian exponential map $\exp\sr$ for the case of 2-step Carnot groups. Other features of normal CC-geodesics are then studied. We show how the system of normal CC-geodesic equations can be integrated step by step. Finally, we show a regularity property of the CC-distance function $\delta\cc$ from a $\cont^k$-smooth hypersurface $S$., Comment: 54 pages
- Published
- 2009
21. Approximation of the Helfrich's functional via Diffuse Interfaces
- Author
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Bellettini, Giovanni and Mugnai, Luca
- Subjects
Mathematics - Analysis of PDEs ,49J45 ,34K26 ,49Q15 ,49Q20 - Abstract
We give a rigorous proof of the approximability of the so-called Helfrich's functional via diffuse interfaces, under a constraint on the ratio between the bending rigidity and the Gauss-rigidity.
- Published
- 2009
22. The regularity of Special Legendrian Integral Cycles
- Author
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Bellettini, Costante and Riviere, Tristan
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,49Q05 ,49Q15 ,32Q25 ,32Q65 - Abstract
Special Legendrian Integral Cycles in $S^5$ are the links of the tangent cones to Special Lagrangian integer multiplicity rectifiable currents in Calabi-Yau 3-folds. We show that such Special Legendrian Cycles are smooth except possibly at isolated points., Comment: 72 pages
- Published
- 2009
23. Leaf superposition property for integer rectifiable currents
- Author
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Ambrosio, Luigi, Crippa, Gianluca, and LeFloch, Philippe G.
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,49Q15 ,26B30 ,30C55 - Abstract
We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation., Comment: 13 pages
- Published
- 2008
24. Ground states in complex bodies
- Author
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Mariano, Paolo Maria and Modica, Giuseppe
- Subjects
Mathematical Physics ,74A30 ,74A35 ,49Q15 ,49J45 ,82B30 - Abstract
A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappinngs and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance of configurational actions follows. After describing a list of possible applications of the general results collected here, a concrete discussion of the existence of ground states in thermodynamically stable quasicrystals is presented at the end., Comment: 30 pages, in print on ESAIM-COCV
- Published
- 2008
25. H\'older forms and integrability of invariant distributions
- Author
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Simić, Slobodan N.
- Subjects
Mathematics - Dynamical Systems ,Mathematics - Classical Analysis and ODEs ,37D30 ,37D10 ,49Q15 - Abstract
We prove an inequality for H\"older continuous differential forms on compact manifolds in which the integral of the form over the boundary of a sufficiently small, smoothly immersed disk is bounded by a certain multiplicative convex combination of the volume of the disk and the area of its boundary. This inequality has natural applications in dynamical systems, where H\"older continuity is ubiquitous. We give two such applications. In the first one, we prove a criterion for the existence of global cross sections to Anosov flows in terms of their expansion-contraction rates. The second application provides an analogous criterion for non-accessibility of partially hyperbolic diffeomorphisms., Comment: The paper has been revised. To appear in Discrete and Continuous Dynamical Systems
- Published
- 2007
26. The Hessian of the distance from a surface in the Heisenberg group
- Author
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Arcozzi, Nicola and Ferrari, Fausto
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Metric Geometry ,49Q15 ,53C17 ,53C22 - Abstract
We compute the horizontal Hessian of the signed Carnot-Charatheodory distance from a surface S in the Heisenberg group H. The expression for the Hessian is in terms of the surface's intrinsic curvatures. As an application, we compute the horizontal Hessian of the Carnot-Charatheodory distance from a point in H.
- Published
- 2006
27. Differential complexes and exterior calculus
- Author
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Harrison, Jenny
- Subjects
Mathematical Physics ,Mathematics - Classical Analysis and ODEs ,Mathematics - Differential Geometry ,49Q15 ,58C35, 15A75, 16E45 - Abstract
In this paper we present a new theory of calculus over $k$-dimensional domains in a smooth $n$-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum with respect to a norm on the space of ``pointed chains,'' culminating in the chainlet complex. Through this complex, we discover a broad theory of coordinate free, multivector analysis in smooth manifolds for which both the classical Newtonian calculus and the Cartan exterior calculus become special cases. The chainlet operators, products and integrals apply to both symmetric and antisymmetric tensor cochains. As corollaries, we obtain the full calculus on Euclidean space, cell complexes, bilayer structures (e.g., soap films) and nonsmooth domains, with equal ease. The power comes from the recently discovered prederivative and preintegral that are antecedent to the Newtonian theory. These lead to new models for the continuum of space and time, and permit analysis of domains that may not be locally Euclidean, or locally connected, or with locally finite mass., Comment: 50 pages
- Published
- 2006
28. Flat convergence for integral currents in metric spaces
- Author
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Wenger, Stefan
- Subjects
Mathematics - Differential Geometry ,49Q15 ,49Q20 - Abstract
It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in particular all Banach spaces as well as complete CAT(k)-spaces (metric spaces of curvature bounded above by k in the sense of Alexandrov). The main result can be used to prove the existence of minimal elements in a fixed Lipschitz homology class in compact metric spaces admitting local cone type inequalities or to conclude that integral currents which are weak limits of sequences of absolutely area minimizing integral currents are again absolutely area minimizing.
- Published
- 2005
29. Uniqueness of tangent cones for calibrated 2-cycles
- Author
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Pumberger, David and Riviere, Tristan
- Subjects
Mathematics - Differential Geometry ,Mathematics - Analysis of PDEs ,49Q15 ,35J60 ,53C38 - Abstract
We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex manifolds and deduce the uniqueness of their tangent maps., Comment: 37 pages
- Published
- 2005
- Full Text
- View/download PDF
30. Ravello lecture notes on geometric calculus -- Part I
- Author
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Harrison, Jenny
- Subjects
Mathematical Physics ,Mathematics - Classical Analysis and ODEs ,49Q15 - Abstract
In these notes of lectures at the 2004 Summer School of Mathematical Physics in Ravello, Italy, the author develops an approach to calculus in which more efficient choices of limits are taken at key points of the development. For example, $k$-dimensional tangent spaces are replaced by representations of simple $k$-vectors supported in single points as limits of simplicial $k$-chains in a Banach space (much like Dirac monopoles). This subtle difference has powerful advantages that will be explored. Through these ``infinitesimals'', we obtain a coordinate free theory on manifolds that builds upon the Cartan exterior calculus. An infinite array of approximating theories to the calculus of Newton and Lebiniz becomes available and one can now revisit old philosophical questions such as which models are most natural for the continuum or for physics. Applications include three extensions of calculus: calculus on fractals, bilayer calculus (soap bubbles) and discrete calculus. This paper is a draft of the first half of the lectures., Comment: 83 pages
- Published
- 2004
31. Estimates and identities for the average distortion of a linear transformation
- Author
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Rivin, Igor
- Subjects
Mathematics - Classical Analysis and ODEs ,Mathematics - Dynamical Systems ,37D25 ,49Q15 - Abstract
For a linear transformation A from Rn to Rn, we give sharp bounds for the average distortion of A, that is, the average value of log of the euclidean norm of Au over all unit vectors u. This is closely related to the results of the author's paper math.DS/0312048., Comment: six pages
- Published
- 2004
32. Geometric Hodge Star Operator with Applications to the Theorems of Gauss and Green
- Author
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Harrison, Jenny
- Subjects
Mathematical Physics ,49Q15 - Abstract
The classical divergence theorem for an $n$-dimensional domain $A$ and a smooth vector field $F$ in $n$-space $$\int_{\partial A} F \cdot n = \int_A div F$$ requires that a normal vector field $n(p)$ be defined a.e. $p \in \partial A$. In this paper we give a new proof and extension of this theorem by replacing $n$ with a limit $\star \partial A$ of 1-dimensional polyhedral chains taken with respect to a norm. The operator $\star$ is a geometric dual to the Hodge star operator and is defined on a large class of $k$-dimensional domains of integration $A$ in $n$-space the author calls {\em chainlets}. Chainlets include a broad range of domains, from smooth manifolds to soap bubbles and fractals. We prove as our main result the Star theorem $$\int_{\star A} \omega = (-1)^{k(n-k)}\int_A \star \omega.$$ When combined with the general Stokes' theorem for chainlet domains $$\int_{\partial A} \omega = \int_A d \omega$$ this result yields optimal and concise forms of Gauss' divergence theorem $$\int_{\star \partial A}\omega = (-1)^{(k-1)(n-k+1)} \int_A d\star \omega$$ and Green's curl theorem $$\int_{\partial A} \omega = \int_{\star A} \star d\omega.$$, Comment: 26 pages, 4 figures
- Published
- 2004
33. On Plateau's Problem for Soap Films with a Bound on Energy
- Author
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Harrison, Jenny
- Subjects
Mathematics - Differential Geometry ,Mathematical Physics ,28A75 ,49Q15 - Abstract
We prove existence and a.e. regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in R^3. The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, integral currents, nonorientable surfaces and soap films as observed by Plateau with a bound on energy. Our area minimizing solution is shown to be a smooth surface away from its branched set which is a union of Lipschitz Jordan curves of finite total length., Comment: 13 pages, 1 figure
- Published
- 2004
34. Cartan's Magic Formula and Soap Film Structures
- Author
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Harrison, Jenny
- Subjects
Mathematics - Classical Analysis and ODEs ,Mathematical Physics ,Mathematics - Geometric Topology ,58K99 ,49Q15 - Abstract
A soap film is actually a thin solid fluid bounded by two surfaces of opposite orientation. It is natural to model the film using one polyhedron for each side. Two problems are to get the polyhedra for both sides to be in the same place without canceling each other out and to model triple junctions without introducing extra boundary components. We use chainlet geometry to create dipole cells and mass cells which accomplish these goals and model faithfully all observable soap films and bubbles. We introduce a new norm on chains of these cells and prove lower semicontinuity of area. A geometric version of Cartan's magic formula provides the necessary boundary coherence., Comment: AMS-LaTeX v1.2, 16 pages with 3 figures; uses birkart.cls
- Published
- 2004
35. The Singular Set of 1-1 Integral Currents
- Author
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Riviere, Tristan and Tian, Gang
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,49Q05 ,49Q15 ,32Q65 - Abstract
We prove that 2 dimensional Integral currents (i.e. integer multiplicity 2 dimensional rectifiable currents) which are almost complex cycles in an almost complex manifold admitting locally a compatible symplectic form are smooth surfaces aside from isolated points and therefore are J-holomorphic curves., Comment: 67 pages
- Published
- 2003
36. Regularity and splitting of directed minimal cones
- Author
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Schnuerer, Oliver C.
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,49Q15 ,35D10 - Abstract
We show that directed minimal cones in (n+1)-dimensional Euclidean space which have at most one singularity are - besides the trivial cases: empty set, whole space - half spaces. Using blow-up techniques, this result can be used to get C^{1,lambda}-regularity for the measure-theoretic boundary of almost minimal Caccioppoli sets which are representable as subgraphs in R^n, n<=8. This provides a different method to obtain a result due to De Giorgi. We also prove a splitting theorem for general directed minimal cones. Such a cone is the Cartesian product of R^k and C, where C is an undirected minimal cone or a half-line., Comment: 20 pages, no figures
- Published
- 2003
37. Isoperimetric inequalities of euclidean type in metric spaces
- Author
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Wenger, Stefan
- Subjects
Mathematics - Functional Analysis ,Mathematics - Metric Geometry ,49Q15 - Abstract
In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in the sense of Alexandrov or, more generally, of Busemann. The main theorem generalizes results of Gromov and Ambrosio-Kirchheim.
- Published
- 2003
38. The min--max construction of minimal surfaces
- Author
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Colding, Tobias H. and De Lellis, Camillo
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Differential Geometry ,Mathematics - Geometric Topology ,49Q05 ,53A10 ,49Q15 - Abstract
In this paper we survey with complete proofs some well--known, but hard to find, results about constructing closed embedded minimal surfaces in a closed 3-dimensional manifold via min--max arguments. This includes results of J. Pitts, F. Smith, and L. Simon and F. Smith., Comment: 42 pages, 13 figures
- Published
- 2003
39. Morse Novikov theory and cohomology with forward supports
- Author
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Harvey, Reese F. and Minervini, G.
- Subjects
Mathematics - Differential Geometry ,Mathematics - Dynamical Systems ,37D15 (Primary) ,49Q15 ,58A12 (Secondary) - Abstract
We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of the manifold. This complex is then used in Novikov theory, to obtain a geometric realization of the Novikov Complex as a complex of currents and a new characterization of Novikov Homology as cohomology with compact forward supports. Two natural ``backward-forward'' dualities are also established: a Lambda duality over the Novikov Ring and a Topological Vector Space duality over the reals., Comment: 34 pages. This version is extended and widely revised. An introduction and many references are added and some corrections are made
- Published
- 2002
40. On a relation between stochastic integration and geometric measure theory
- Author
-
Flandoli, Franco, Giaquinta, Mariano, Gubinelli, Massimiliano, and Tortorelli, Vincenzo M.
- Subjects
Mathematics - Probability ,60H05 ,49Q15 - Abstract
Two problems are addressed for the path of certain stochastic processes: a) do they define currents? b) are these currents of a classical type? A general answer to question a) is given for processes like semimartingales or with Lyons-Zheng structure. As to question b), it is shown that H\"{o}lder continuous paths with exponent $\gamma > 1/2$ define integral flat chains., Comment: 30 pages, no figures
- Published
- 2002
41. Projecting (n-1)-cycles to zero on hyperplanes in R^{n+1}
- Author
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Solomon, Bruce
- Subjects
Mathematics - Differential Geometry ,53A07 ,53C42 ,49Q15 - Abstract
The projection of a compact oriented submanifold M^{n-1} in R^{n+1} on a hyperplane P^{n} can fail to bound any region in P. We call this ``projecting to zero.'' Example: The equatorial S^1 in S^2 projects to zero in any plane containing the x_3-axis. Using currents to make this precise, we show: A lipschitz (homology) (n-1)-sphere embedded in a compact, strictly convex hypersurface cannot project to zero on n+1 linearly independent hyperplanes in R^{n+1}. We also show, using examples, that all the hypotheses in this statement are sharp., Comment: 14 pages, 4 figures
- Published
- 2002
42. Hyperbolic Unfoldings of Minimal Hypersurfaces
- Author
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Lohkamp Joachim
- Subjects
singularities ,uniform spaces ,gromov hyperbolicity ,bounded geometry ,minimal hypersurfaces ,s-structures ,conformal deformations ,30l99 ,51m10 ,49q15 ,53a10 ,53a30 ,Analysis ,QA299.6-433 - Abstract
We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.
- Published
- 2018
- Full Text
- View/download PDF
43. Poincaré-type inequalities and finding good parameterizations.
- Author
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Merhej, Jessica
- Abstract
A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. Reifenberg (Bull Am Math Soc 66:312–313, 1960) proved that if a set is well approximated by planes at every point and at every scale, then the set is a bi-Hölder image of a plane. It is known today that Carleson-type conditions on these approximating planes guarantee a bi-Lipschitz parameterization of the set. In this paper, we consider an n-Ahlfors regular rectifiable set M ⊂ R n + d that satisfies a Poincaré-type inequality involving Lipschitz functions and their tangential derivatives. Then, we show that a Carleson-type condition on the oscillations of the tangent planes of M guarantees that M is contained in a bi-Lipschitz image of an n-plane. We also explore the Poincaré-type inequality considered here and show that it is in fact equivalent to other Poincaré-type inequalities considered on general metric measure spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
44. Multiple valued sections of vector bundles: the reparametrization theorem for $Q$-valued functions revisited
- Author
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Stuvard, Salvatore
- Subjects
Mathematics - Differential Geometry ,Statistics and Probability ,49Q15 ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,FOS: Mathematics ,Geometry and Topology ,Statistics, Probability and Uncertainty ,Analysis ,Analysis of PDEs (math.AP) - Abstract
We analyze a notion of multiple valued sections of a vector bundle over an abstract smooth Riemannian manifold, which was suggested by W. Allard in the unpublished note "Some useful techniques for dealing with multiple valued functions" and generalizes Almgren's $Q$-valued functions. We study some relevant properties of such $Q$-multisections and apply the theory to provide an elementary and purely geometric proof of a delicate reparametrization theorem for multi-valued graphs which plays an important role in the regularity theory for higher codimension area minimizing currents \`a la Almgren-De Lellis-Spadaro., Comment: V2 is the final version, to appear in Comm. Anal. Geom
- Published
- 2022
45. Quantitative minimality of strictly stable extremal submanifolds in a flat neighbourhood.
- Author
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Inauen, Dominik and Marchese, Andrea
- Subjects
- *
SUBMANIFOLDS , *GEOMETRIC measure theory , *CURRENTS (Calculus of variations) , *ISOPERIMETRIC inequalities , *PROBLEM solving - Abstract
In this paper we extend the results of A strong minimax property of nondegenerate minimal submanifolds , by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood. We prove a similar result, replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates. Our proof is based on the introduction of a penalized minimization problem, in the spirit of A selection principle for the sharp quantitative isoperimetric inequality , by Cicalese and Leonardi, which allows us to exploit the regularity theory for almost minimizers of elliptic parametric integrands. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
46. Unique tangent behavior for 1-dimensional stationary varifolds.
- Author
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Liang, Xiangyu
- Subjects
- *
VARIFOLDS , *TANGENTS (Geometry) , *RIEMANNIAN manifolds , *MATHEMATICAL bounds , *MATHEMATICAL analysis - Abstract
We prove that, without any assumption on lower density bound or codimension, any 1-dimensional stationary varifold on any Riemannian manifold admits unique tangent behavior everywhere. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
47. Une version quasi-symétrique du problème d'équivalence höldérienne pour les groupes de Carnot
- Author
-
Pierre Pansu, Université Paris Sud, ANR-10-BLAN-0116,GGAA,Aspects geometriques, analytiques et algorithmiques des groupes(2010), and European Project: 607643,EC:FP7:PEOPLE,FP7-PEOPLE-2013-ITN,MANET(2014)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Hölder condition ,Riemannian geometry ,Heisenberg group ,01 natural sciences ,symbols.namesake ,Mathematics - Metric Geometry ,0103 physical sciences ,Mathematics::Metric Geometry ,0101 mathematics ,[MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG] ,Equivalence (measure theory) ,Mathematics ,Hölder continuity ,packing measure ,010102 general mathematics ,49Q15 ,30L10 ,30F45 ,43A80 ,22E46 ,General Medicine ,curvature pinching ,symmetric space ,coarea formula ,quasisymmetric ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Symmetric space ,symbols ,Coarea formula ,Mathematics::Differential Geometry ,010307 mathematical physics ,Carnot cycle - Abstract
A variant of Gromov's Hölder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.; On introduit une variante, invariante par homéomorphisme quasi-symétrique, du problème d'équivalence höldérienne de Gromov. On obtient un résultat partiel, qui a une conséquence en géométrie riemannienne. Il repose sur une forme générale de l'inégalité de la coaire pour les p-énergies des fonctions.
- Published
- 2020
48. Computing Gaussian & exponential measures of semi-algebraic sets.
- Author
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Lasserre, Jean B.
- Subjects
- *
GAUSSIAN function , *EXPONENTIAL functions , *SET theory , *CARLEMAN theorem , *MATHEMATICAL bounds - Abstract
We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure μ ( Ω ) of (not necessarily compact) basic semi-algebraic sets Ω ⊂ R n . We obtain two monotone (non-increasing and non-decreasing) sequences of upper and lower bounds ( ω ‾ d ) , ( ω _ d ) , d ∈ N , each converging to μ ( Ω ) as d → ∞ . For each d , computing ω ‾ d or ω _ d reduces to solving a semidefinite program whose size increases with d . Some preliminary (small dimension) computational experiments are encouraging and illustrate the potential of the method. The method also works for any measure whose moments are known and which satisfies Carleman's condition. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
49. The resolution of the Yang–Mills Plateau problem in super-critical dimensions.
- Author
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Petrache, Mircea and Rivière, Tristan
- Subjects
- *
YANG-Mills theory , *DIMENSION theory (Algebra) , *FUNCTION spaces , *CURRENTS (Calculus of variations) , *SHEAF theory - Abstract
We study the minimization problem for the Yang–Mills energy under fixed boundary connection in supercritical dimension n ≥ 5 . We define the natural function space A G in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A G can be also interpreted as a space of weak connections on a “real measure theoretic version” of reflexive sheaves from complex geometry. We prove the existence of weak solutions to the Yang–Mills Plateau problem in the space A G . We then prove the optimal regularity result for solutions of this Plateau problem. On the way to prove this result we establish a Coulomb gauge extraction theorem for weak curvatures with small Yang–Mills density. This generalizes to the general framework of weak L 2 curvatures previous works of Meyer–Rivière and Tao–Tian in which respectively a strong approximability property and an admissibility property were assumed in addition. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
50. On extending calibration pairs.
- Author
-
Zhang, Yongsheng
- Subjects
- *
CALIBRATION , *SUBMANIFOLDS , *RIEMANNIAN manifolds , *SET theory , *MATHEMATICAL singularities - Abstract
The paper studies how to extend local calibration pairs to global ones in various situations. As a result, new discoveries involving mass-minimizing properties are exhibited. In particular, we show that a R -homologically nontrivial connected submanifold M of a smooth Riemannian manifold X is homologically mass-minimizing for some metrics in the same conformal class. Moreover, several generalizations for M with multiple connected components or for a mutually disjoint collection (see § 3.5 ) are obtained. For a submanifold with certain singularities, we also establish an extension theorem for generating global calibration pairs. By combining these results, we find that, in some Riemannian manifolds, there are homologically mass-minimizing smooth submanifolds which cannot be calibrated by any smooth calibration. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
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