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In the context of synthetic differential geometry, we study the Laplace operator an a Riemannian manifold. The main newaspect is a neighbourhood of the diagonal, smaller than the second neighbourhood usually required as support for second order differential operators. The newneighbourhood has the property that a function is affine on it if and only if it is harmonic.

We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

In this paper, we study the expansions of Ricci flat metrics in harmonic coordinates about the infinity of ALE (asymptotically local Euclidean) manifolds., 23 pages

In this paper, we prove the existence of $H^2$-regular coordinates on Riemannian $3$-manifolds with boundary, assuming only $L^2$-bounds on the Ricci curvature, $L^4$-bounds on the second fundamental form of the boundary, and a positive lower bound on the volume radius. The proof follows by extending the theory of Cheeger-Gromov convergence to include manifolds with boundary in the above low regularity setting. The main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate, and a rigidity argument using manifold doubling. Assuming higher regularity of the Ricci curvature, we also prove corresponding higher regularity estimates for the coordinates., Comment: 44 pages; part 1 of a revised version of "Boundary harmonic coordinates and the localised bounded $L^2$-curvature theorem". All comments welcome!

Barycentric coordinates provide a convenient way to represent a point inside a triangle as a convex combination of the triangle’s vertices and to linearly interpolate data given at these vertices. Due to their favourable properties, they are commonly applied in geometric modelling, finite element methods, computer graphics, and many other fields. In some of these applications, it is desirable to extend the concept of barycentric coordinates from triangles to polygons, and several variants of such generalized barycentric coordinates have been proposed in recent years. In this paper we focus on exponential three-point coordinates, a particular one-parameter family for convex polygons, which contains Wachspress, mean value, and discrete harmonic coordinates as special cases. We analyse the behaviour of these coordinates and show that the whole family is C 0 at the vertices of the polygon and C 1 for a wide parameter range.

Coordinate invariance of physical laws is central in physics, it grants us the freedom to express observations in coordinate systems that provide computational convenience. In the context of medical imaging there are numerous examples where departing from Cartesian to curvilinear coordinates leads to ease of visualization and simplicity, such as spherical coordinates in the brain's cortex, or universal ventricular coordinates in the heart. In this work we introduce tools that enhance the use of existing diffusion tractography approaches to utilize arbitrary coordinates. To test our method we perform simulations that gauge tractography performance by calculating the specificity and sensitivity of tracts generated from curvilinear coordinates in comparison with those generated from Cartesian coordinates, and we find that curvilinear coordinates generally show improved sensitivity and specificity compared to Cartesian. Also, as an application of our method, we show how harmonic coordinates can be used to enhance tractography for the hippocampus.

The need for more and more accurate gravitational wave templates requires taking into account all possible contributions to the emission of gravitational radiation from a binary system. Therefore, working within a multipolar-post-Minkowskian framework to describe the gravitational wave field in terms of the source multipole moments, the dominant instantaneous effects should be supplemented by hereditary contributions arising from nonlinear interactions between the multipoles. The latter effects are referred to as tails being described in terms of integrals depending on the past history of the source. We compute higher-order tail (i.e., tail-of-tail and tail-squared) contributions to both energy and angular momentum fluxes and their averaged values along hyperboliclike orbits at the leading post-Newtonian approximation, using harmonic coordinates and working in the Fourier domain. Due to the increasing level of accuracy recently achieved in the determination of the scattering angle in a two-body system by several complementary approaches, the knowledge of these terms will provide useful information to compare results from different formalisms., Comment: 14 pages, 1 figure (2 eps files), revtex macros used; v2: Eq. (4.14) corrected

In previous works, we obtained the leading, next-to-leading and next-to-next-to-leading (NNL) post-Newtonian (PN) corrections in the conservative tidal interactions between two compact non-spinning objects using a Lagrangian of effective field theory (EFT) in harmonic coordinates. In the present paper, we compute the corresponding NNL PN tidal effective Hamiltonian in ADM-like and isotropic coordinates, with contributions from mass quadrupole, current quadrupole and mass octupole tidal interactions, consistently included at that level. We also derive the NNL tidal Hamiltonian in Delaunay variables. We find full agreement in the overlap with recent results that were derived using tools from scattering amplitudes and the EFT to second post-Minkowskian (PM) order., 10 pages

In some simple models of magnetospheric electrodynamics [e.g., Volland, Ann. Geophys., 31, 154–173, 1975], the normal component of the convection electric field is discontinuous across the boundary between closed and open magnetic field lines, and this discontinuity facilitates the formation of auroral arcs there. The requisite discontinuity in E is achieved by making the scalar potential proportional to a positive power (typically 1 or 2) of L on closed field lines and to a negative power (typically −1/2) of L on open (i.e., polar-cap) field lines. This suggests it may be advantageous to construct more realistic (and thus more complicated) empirical magnetospheric and ionospheric electric-field models from superpositions of mutually orthogonal (or not) vector basis functions having this same analytical property (i.e., discontinuity at L = L*, the boundary surface between closed and open magnetic field lines). The present work offers a few examples of ways to make such constructions. A major challenge in this project has been to devise a coordinate system that simplifies the required analytical expansions of electric scalar potentials and accommodates the anti-sunward offset of each polar cap's centroid relative to the corresponding magnetic pole. For circular northern and southern polar caps containing equal amounts of magnetic flux, one can imagine a geometrical construction of coordinate contours such that arcs of great circles connect points of equal quasi-longitude (analogous to MLT) on the northern and southern polar-cap boundaries. For more general polar-cap shapes and (in any case) to assure mutual orthogonality of respective coordinate surfaces on the ionosphere, a formulation based on harmonic coordinate functions (expanded in solutions of the two-dimensional Laplace equation) may be preferable.

Under the usual condition that the volume of a geodesic ball is close to the Euclidean one or the injectivity radii is bounded from below, we prove a lower bound of the $$C^{\alpha } \cap W^{1, q}$$ harmonic radius for manifolds with bounded Bakry–Emery Ricci curvature when the gradient of the potential is bounded. Under these conditions, the regularity that can be imposed on the metrics under harmonic coordinates is only $$C^\alpha \cap W^{1,q}$$ , where $$q>2n$$ and n is the dimension of the manifolds. This is almost 1-order lower than that in the classical $$C^{1,\alpha } \cap W^{2, p}$$ harmonic coordinates under bounded Ricci curvature condition (Anderson in Invent Math 102:429–445, 1990). The loss of regularity induces some difference in the method of proof, which can also be used to address the detail of $$W^{2, p}$$ convergence in the classical case. Based on this lower bound and the techniques in Cheeger and Naber (Ann Math 182:1093–1165, 2015) and Wang and Zhu (Crelle’s J, http://arxiv.org/abs/1304.4490 ), we extend Cheeger–Naber’s Codimension 4 Theorem in Cheeger and Naber (2015) to the case where the manifolds have bounded Bakry–Emery Ricci curvature when the gradient of the potential is bounded. This result covers Ricci solitons when the gradient of the potential is bounded. During the proof, we will use a Green’s function argument and adopt a linear algebra argument in Bamler (J Funct Anal 272(6):2504–2627, 2017). A new ingredient is to show that the diagonal entries of the matrices in the Transformation Theorem are bounded away from 0. Together these seem to simplify the proof of the Codimension 4 Theorem, even in the case where Ricci curvature is bounded.

In the present work, the harmonic vector field is defined on closed Finsler measure spaces through different approaches. At first, the weighted harmonic vector field is obtained as the solution space of a PDE system. Then a suitable Dirichlet energy functional is introduced. A σ-harmonic vector field is considered as the critical point of related action. It is proved that a σ-harmonic vector field on a closed Finsler space with an extra unit norm condition is an eigenvector of the defined Laplacian operator on vector fields. Moreover, we prove that a unit weighted harmonic vector field on a closed generalized Einstein manifold is a σ-harmonic vector field.

A growth of harmonic functions in the whole space ℝn is examined. We found the estimate for a uniform norm of spherical harmonics in terms of the best approximation of harmonic function in the ball by harmonic polynomials. An approximation error of harmonic function in the ball is estimated by the maximum modulus of an entire harmonic function in space, as well as the maximum modulus of an entire harmonic function in space in terms of the maximum modulus of some entire function of one complex variable or the maximal term of its power series. These results allowed us to obtain the necessary and sufficient conditions under which a harmonic function in the ball of an n-dimensional space, n≥3, can be continued to the entire harmonic one. This result is formulated in terms of the best approximation of the given function by harmonic polynomials. In order to characterize growth of an entire harmonic function, we used the generalized and the lower generalized orders. Formulae for the generalized and the lower generalized orders of an entire harmonic function in space are expressed in terms of the approximation error by harmonic polynomials of the function that continues. We also investigated the growth of functions of slow increase. The obtained results are analogues to classical results, which are known for the entire functions of one complex variable.The conducted research is important due to the fact that the harmonic functions occupy a special place not only in many mathematical studies, but also when applying mathematical analysis to physics and mechanics, where these functions are often employed to describe various stationary processes

We propose an iteratively adaptive Multi-scale Finite Element Method (MsFEM) for elliptic PDEs with rough coefficients. The choice of the local boundary conditions for the multi-sale basis functions determines the accuracy of the MsFEM numerical solution, and one needs to incorporate the global information of the elliptic equation into the local boundary conditions of the multi-scale basis functions to recover the underlying fine-mesh solution of the equation. In our proposed iteratively adaptive method, we achieve this global-to-local information transfer through the combination of coarse-mesh solving using adaptive multi-scale basis functions and fine-mesh smoothing operations. In each iteration step, we first update the multi-scale basis functions based on the approximate numerical solutions of the previous iteration steps, and obtain the coarse-mesh approximate solution using a Galerkin projection. Then we apply several steps of smoothing operations to the coarse-mesh approximate solution on the underlying fine mesh to get the updated approximate numerical solution. The proposed algorithm can be viewed as a nonlinear two-level multi-grid method with the restriction and prolongation operators adapted to the approximate numerical solutions of the previous iteration steps. Convergence analysis of the proposed algorithm is carried out under the framework of two-level multi-grid method, and the harmonic coordinates are employed to establish the approximation property of the adaptive multi-scale basis functions. We demonstrate the efficiency of our proposed multi-scale methods through several numerical examples including a multi-scale coefficient problem, a high-contrast interface problem, and a convection-dominated diffusion problem.

We present a classification of complete locally irreducible Riemannian manifolds with nonnegative curvature operator, which admit a nonzero and nondecomposable harmonic form with its square-integrable norm. We prove a vanishing theorem for harmonic forms on complete generic Riemannian manifolds with nonnegative curvature operator. We obtain similar results for closed and co-closed conformal Killing forms.

We develop new high-order results for the post-Newtonian (PN) expansions of the energy and angular momentum fluxes at infinity for eccentric-orbit extreme-mass-ratio inspirals (EMRIs) on a Schwarzschild background. The series are derived through direct expansion of the MST solutions within the RWZ formalism for first-order black hole perturbation theory (BHPT). By utilizing factorization and a few computational simplifications, we are able to compute the fluxes to 19PN, with each PN term calculated as a power series in (Darwin) eccentricity to $e^{10}$. This compares favorably with the numeric fitting approach used in previous work. We also compute PN terms to $e^{20}$ through 10PN. Then, we analyze the convergence properties of the composite energy flux expansion by checking against numeric data for several orbits, both for the full flux and also for the individual 220 mode, with various resummation schemes tried for each. The match between the high-order series and numerical calculations is generally strong, maintaining relative error better than $10^{-5}$ except when $p$ (the semi-latus rectum) is small and $e$ is large. However, the full-flux expansion demonstrates superior fidelity (particularly at high $e$), as it is able to incorporate additional information from PN theory. For the orbit $(p=10, e=1/2)$, the full flux achieves a best error near $10^{-5}$, while the 220 mode exhibits error worse than $1\%$. Finally, we describe a procedure for transforming these expansions to the harmonic gauge of PN theory by analyzing Schwarzschild geodesic motion in harmonic coordinates. This will facilitate future comparisons between BHPT and PN theory., Comment: 29 pages, 3 figures

We study the chiral self-gravitating model (CSGM) of a special type in the spherically symmetric static spacetime in Einstein frame. Such CSGM is derived, by virtue of Weyl conformal transformation, from a gravity model in the Jordan frame corresponding to a modified $f(R)$ gravity with a kinetic scalar curvature. We investigate the model using harmonic coordinates and consider a special case of the scaling transformation from the Jordan frame. We find classes of solutions corresponding to a zero potential and we investigate horizons, centers and the asymptotic behavior of the obtained solutions. Other classes of solutions (for the potential not equal to zero) are found using a special relation (ansatz) between the metric components. Investigations of horizons, centers and asymptotic behavior of obtained solutions for this new case are performed as well. Comparative analysis with the solutions obtained earlier in literature is made., 25 pages. arXiv admin note: text overlap with arXiv:2005.11858

We calculate the potential contributions of the Hamiltonian in harmonic coordinates up 6PN for binary mass systems to O ( G N 4 ) and perform comparisons to recent results in the literature [1] and [2] .

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds.

Harmonic coordinates are widely considered to be perfect barycentric coordinates of a polygonal domain due to their attractive mathematical properties. Alas, they have no closed form in general, so must be numerically approximated by solving a large linear equation on a discretization of the domain. The alternatives are a number of other simpler schemes which have closed forms, many designed as a (computationally) cheap approximation to harmonic coordinates. One test of the quality of the approximation is whether the coordinates coincide with the harmonic coordinates for the special case where the polygon is close to a circle (where the harmonic coordinates have a closed form - the celebrated Poisson kernel). Coordinates which pass this test are called "pseudo-harmonic". Another test is how small the differences between the coordinates and the harmonic coordinates are for "real-world" polygons using some natural distance measures.We provide a qualitative and quantitative comparison of a number of popular barycentric coordinate methods. In particular, we study how good an approximation they are to harmonic coordinates. We pay special attention to the Moving-Least-Squares coordinates, provide a closed form for them and their transfinite counterpart (i.e. when the polygon converges to a smooth continuous curve), prove that they are pseudo-harmonic and demonstrate experimentally that they provide a superior approximation to harmonic coordinates. A qualitative and quantitative comparison of popular barycentric coordinate methods.A closed form for MLS coordinates and their transfinite counterpart.Prove MLS coordinates are pseudo-harmonic.Demonstrate MLS coordinates provide a superior approximation to harmonic coordinates.

In this paper, we have studied harmonic maps on trans-Sasakian manifolds. First it is proved that if F: M1 → M2 is a Riemannian ϕ-holomorphic map between two trans-Sasakian manifolds such that ξ2 ∈ (Im dF)⊥, then F can not be harmonic provided that β2 ≠ 0. We have also found the necessary and sufficient condition for the harmonic map to be constant map from Kaehler to trans-Sasakian manifold. Finally, we prove the non-existence of harmonic map from locally conformal Kaehler manifold to trans-Sasakian manifold.

In this paper, we derive a local Hamilton type gradient estimate for a nonlinear heat equation on Riemannian manifolds. As its application, we obtain a Liouville type theorem.

Starting from the Kerr–Schild coordinates, a set of horizon penetrating harmonic coordinates for the Kerr metric is worked out and exhibit the explicit form of the whole metric. As a validity check on the calculations, the multipole expansion of the Kerr metric is then derived from the harmonic formulation. The resemblance of the geometry of the harmonic slicing with the ‘1 + log’ slicing and the generalized harmonic gauge in numerical relativity is then further discussed. Together with its horizon penetrating structure, the harmonic formulation is well suited for the numerical simulation of astrophysical electromagnetic phenomena, like for instance the Blanford–Znajek process, in the vicinity of a supermassive black hole.

Physics letters / B 807, 135496 - (2020). doi:10.1016/j.physletb.2020.135496, We calculate the motion of binary mass systems in gravity up to the sixth post–Newtonian order to the GN3 terms ab initio using momentum expansions within an effective field theory approach based on Feynman amplitudes in harmonic coordinates. For these contributions we construct a canonical transformation to isotropic and to EOB coordinates at 5PN and agree with the results in the literature [1,2] by Bern et al. and Damour. At 6PN we compare to the Hamiltonians in isotropic coordinates either given in [1] or resulting from the scattering angle. We find a canonical transformation from our Hamiltonian in harmonic coordinates to [1] , but not to [2] . This implies that we also agree on all observables with [1] to the sixth post–Newtonian order to GN3 ., Published by North-Holland Publ., Amsterdam

The Multipolar Post-Minkowskian (MPM) formalism represents an approach for determining the metric density in the exterior of a compact source of matter. In the MPM formalism the metric density is given in harmonic coordinates and in terms of symmetric tracefree (STF) multipoles. In this investigation, the post-linear metric density of this formalism is used in order to determine the post-linear metric tensor in the exterior of a compact source of matter. The metric tensor is given in harmonic coordinates and in terms of STF multipoles. The post-linear metric coefficients are associated with an integration procedure. The integration of these post-linear metric coefficients is performed explicitly for the case of a stationary source, where the first multipoles (monopole and quadrupole) of the source are taken into account. These studies are a requirement for further investigations in the theory of light propagation aiming at highly precise astrometric measurements in the solar system, where the post-linear coefficients of the metric tensor of solar system bodies become relevant., Comment: 32 pages, 2 figures

New elements are proposed in the present work based on the harmonic coordinates in which the shape functions satisfy the Laplace equation. The harmonic functions have some appealing characteristics that made it possible to define elements with arbitrary shape functions on the element boundaries and arbitrary node arrangement. In the present work, without loss of generality, we formulate a set of quadrilateral elements with midside nodes and piecewise linear shape functions along the element boundaries. The analytical solution of Laplace equation in rectangular domains are used here to represent the harmonic shape function as Fourier series. To derive the shape functions of different elements with different node arrangements a systematic approach is proposed to represent all of the shape functions using a small set of harmonic functions. Two model boundary value problems, Poisson equation and linear elasticity are used here to evaluate the proposed method. Patch test and convergence analysis are done for each model problems via some examples. The elements with midside nodes have great potential to be used in mesh adaptive solutions. Therefore, some adaptive mesh refinement examples are solved based on the ZZ error estimation and quadtree grid structure as the mesh refinement technique.

We derive from first principles a third post-Newtonian (3PN) accurate Keplerian-type parametric solution to describe PN accurate dynamics of nonspinning compact binaries in hyperbolic orbits. Orbital elements and functions of the parametric solution are obtained in terms of the conserved orbital energy and angular momentum in both Arnowitt-Deser-Misner-type and modified harmonic coordinates. Elegant checks are provided that include a modified analytic continuation prescription to obtain our independent hyperbolic parametric solution from its eccentric version. A prescription to model gravitational wave polarization states for hyperbolic compact binaries experiencing 3.5PN accurate orbital motion is presented that employs our 3PN accurate parametric solution.

Let $(W,q, \mathcal{D})$ be a 4-dimensional Walker manifold. After providing a characterization and some examples for several special $(1,1)$-tensor fields on $(W,q, \mathcal{D})$, we prove that the proper almost complex structure $J$, introduced by Matsushita, is harmonic in the sense of Garcia-Rio et al. if and only if the almost Hermitian structure $(J,q)$ is almost Kahler. We classify all harmonic functions locally defined on $(W,q, \mathcal{D})$. We deal with the harmonicity of quadratic maps defined on $\mathbb{R}^4$ (endowed with a Walker metric $q$) to the $n$-dimensional semi-Euclidean space of index $r$, and then between local charts of two 4-dimensional Walker manifolds. We obtain here the necessary and sufficient conditions under which these maps are harmonic, horizontally weakly conformal, or harmonic morphisms with respect to $q$.

Let (M, g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that if M is asymptotically harmonic of constant h = 0, then M is a flat manifold. This theorem shows that any asymptotically harmonic manifold in dimension 3 is a symmetric space, thus completing the classification of asymptotically harmonic manifolds in dimension 3.

We prove the harmonic map version of the Royden decomposition in the sense that given any bounded C1-map f with finite total energy on a complete Riemannian manifold into a Cartan-Hadamard manifold, there exists a unique bounded harmonic map with finite total energy from the manifold into the Cartan-Hadamard manifold taking the same boundary value at each harmonic boundary point as that of f.

We define two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kahler manifold . As a consequence we can construct sequences of H-surfaces and almost complex surfaces.

We show that under gravitational contraction of a spherical body, its internal energy increases infinitely as the body radius approaches GM/c 2 , which leads to a negative mass defect and unavoidably to an explosion process with ejection of part of the body mass because the spherical compression is unstable. This conclusion follows exactly from the general theory of relativity in harmonic coordinates.

By using potential-theoretic methods on locally compact spaces, two harmonic sheaves (resembling the solutions of the Laplace and the Schrodinger operators) on an infinite network are compared.

While the equations of general relativity take the same form in any coordinate system, choosing a suitable set of coordinates is essential in any practical application. This poses a challenge in background-independent quantum gravity, where coordinates are not a priori available and need to be reconstructed from physical degrees of freedom. We review the general idea of coupling free scalar fields to gravity and using these scalars as a "matter reference frame." The resulting coordinate system is harmonic, i.e. it satisfies harmonic (de Donder) gauge. We then show how to introduce such matter reference frames in the group field theory approach to quantum gravity, where spacetime is emergent from a "condensate" of fundamental quantum degrees of freedom of geometry, and how to use matter coordinates to extract physics. We review recent results in homogeneous and inhomogeneous cosmology, and give a new application to the case of spherical symmetry. We find tentative evidence that spherically symmetric group field theory condensates defined in this setting can reproduce the near-horizon geometry of a Schwarzschild black hole., Comment: 18 pages, revtex; v2: expanded discussion, added references, new title, no change in results

In this paper, we propose a multiscale data-driven stochastic method (MsDSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. This method combines the advantages of the recently developed multiscale model reduction method [M. L. Ci, T. Y. Hou, and Z. Shi, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 449--474] and the data-driven stochastic method (DSM) [M. L. Cheng et al., SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 452--493]. Our method consists of offline and online stages. In the offline stage, we decompose the harmonic coordinate into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Based on the Karhunen--Loève (KL) expansion of the smooth parts and oscillatory parts of the harmonic coordinates, we can derive an effective stochastic equation that can be well-resolved on a coarse grid. We then apply the DSM to the effective stochastic equation to construct a data-driven stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions. In the online stage, we expand the SPDE solution using the data-driven stochastic basis and solve a small number of coupled deterministic partial differential equations (PDEs) to obtain the expansion coefficients. The MsDSM reduces both the stochastic and the physical dimensions of the solution. We have performed complexity analysis which shows that the MsDSM offers considerable savings over not only traditional methods but also DSM in solving multiscale SPDEs. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.

This chapter reviews some important properties of Laplacians, smooth and discrete. The Laplacian is perhaps the prototypical differential operator for various physical phenomena. The properties mentioned so far play an important role in applications; specifically, in the context of barycentric coordinates, they give rise to mean value coordinates and harmonic coordinates. Discrete Laplacians can be defined on simplicial manifolds or, more generally, on graphs. Positive edge weights are a natural choice if weights resemble transition probabilities of a random walker. Discrete Laplacians with positive weights are always positive semidefinite (Psd) and, just like in the smooth setting, they only have the constant functions in their kernel provided that the graph is connected. The construction of discrete Laplacians based on inner products can be extended from simplicial surfaces to meshes with (not necessarily planar) polygonal faces; for such an extension that is similar to the approach considered for planar polygons.

We say that a distribution is harmonic if it is harmonic when considered as a section of a Grassmann bundle. We find new examples of harmonic distributions and show nonexistense of harmonic distrubutions on some Riemannian manifolds by two different approaches. Firstly, we lift distributions to the second tangent bundle equipped with the Sasaki metric. Secondly, we deform conformally the metric on a base manifold., Comment: Revised version of the previous preprint 'Harmonic distributions and conformal deformations', 13 pages

The present paper establishes connections between various subclasses of harmonic univalent functions by applying certain integral operator involving the generalized Bessel functions of first kind. To be more precise, we investigate such connections with harmonic starlike and harmonic convex mappings in the plane.

For the harmonic function on two dimensional Riemannian manifolds with constant Gaussian curvature, some differential equalities and inequalities are established to study the curvature estimate of the steepest descents by the maximum principle.

In this paper, we consider the harmonic maps from a Riemannian manifold with non-positive pinching curvature to any Finsler manifold, and we can prove that there is no non-degenerate harmonic maps from a classical bounded symmetric domain to any Finsler manifold with moderate divergent energy. In particular, we obtain that any harmonic map from a classical bounded symmetric domain to any Riemannian manifold with finite energy has to be constant, which improves the Xin’s result in Acta Math Sinica 15:277–292, 1999.

We prove that given any continuous data f on the harmonic boundary of a complete Riemannian manifold with image within a ball in the normal range, there exists a harmonic map from the manifold into the ball taking the same boundary value at each harmonic boundary point as that of f.

In this paper, we present the Multiscale Finite Element Method (MsFEM) for problems on rough heterogeneous surfaces. We consider the diffusion equation on oscillatory surfaces. Our objective is to represent small-scale features of the solution via multiscale basis functions described on a coarse grid. This problem arises in many applications where processes occur on surfaces or thin layers. We present a unified multiscale finite element framework that entails the use of transformations that map the reference surface to the deformed surface. The main ingredients of MsFEM are (1) the construction of multiscale basis functions and (2) a global coupling of these basis functions. For the construction of multiscale basis functions, our approach uses the transformation of the reference surface to a deformed surface. On the deformed surface, multiscale basis functions are defined where reduced (1D) problems are solved along the edges of coarse-grid blocks to calculate nodal multiscale basis functions. Furthermore, these basis functions are transformed back to the reference configuration. We discuss the use of appropriate transformation operators that improve the accuracy of the method. The method has an optimal convergence if the transformed surface is smooth and the image of the coarse partition in the reference configuration forms a quasiuniform partition. In this paper, we consider such transformations based on harmonic coordinates (following H. Owhadi and L. Zhang [Comm. Pure and Applied Math., LX(2007), pp. 675-723]) and discuss gridding issues in the reference configuration. Numerical results are presented where we compare the MsFEM when two types of deformations are used for multiscale basis construction. The first deformation employs local information and the second deformation employs a global information. Our numerical results show that one can improve the accuracy of the simulations when a global information is used.

Imaging and simulation methods are typically constrained to resolutions much coarser than the scale of physical micro-structures present in body tissues or geological features. Mathematical and numerical homogenization address this practical issue by identifying and computing appropriate spatial averages that result in accuracy and consistency between the macro-scales we observe and the underlying micro-scale models we assume. Among the various applications benefiting from homogenization, Electric Impedance Tomography (EIT) images the electrical conductivity of a body by measuring electrical potentials consequential to electric currents applied to the exterior of the body. EIT is routinely used in breast cancer detection and cardio-pulmonary imaging, where current flow in fine-scale tissues underlies the resulting coarse-scale images. In this paper, we introduce a geometric approach for the homogenization (simulation) and inverse homogenization (imaging) of divergenceform elliptic operators with rough conductivity coefficients in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions over the domain. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of the domain when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. We explicitly give the transformations which map conductivity coefficients into divergencefree matrices and convex functions, as well as their respective inverses. Using weighted Delaunay triangulations for linearly interpolating convex functions, we apply this geometric framework to obtain a robust homogenization algorithm for arbitrary rough coefficients, extending the global optimality of Delaunay triangulations with respect to a discrete Dirichlet energy to weighted Delaunay triangulations. We then consider inverse homogenization, which is known to be both non-linear and severely illposed, but that we can decompose into a linear ill-posed problem and a well-posed non-linear problem. Finally, our new geometric approach to homogenization and inverse homogenization is applied to EIT.

The notion of exponentially harmonic maps was introduced by Eells and Lemaire (Proceedings of the Banach Center Semester on PDE, pp. 1990–1991, 1990). In this note, by using the maximum principle we get the gradient estimate of exponentially harmonic functions, and then derive a Liouville type theorem for bounded exponentially harmonic functions on a complete Riemannian manifold with nonnegative Ricci curvature and sectional curvature bounded below.

This study is concerned with univalent harmonic mappings on the unit disc of the complex plane. We examine a family of harmonic polynomials that have some interesting geometric properties, e.g. the boundary curve is concave except for cusps, that is dense in the family of harmonic mappings.

We prove that the distortion function of the Gauss map of a surface parametrized by harmonic coordinates coincides with the distortion function of the parametrization. Consequently, the Gauss map of a harmonic surface is K quasiconformal if and only if its harmonic parametrization is K quasiconformal, provided that the Gauss map is regular or what is shown to be the same, provided that the surface is non-planar. This generalizes the classical result that the Gauss map of a minimal surface is a conformal mapping.

Given an open hyperbolic Riemannian manifold, we show that various vector spaces of harmonic functions coincide if and only if they are finite dimensional.

The main general subject of this chapter is an inverse problem of identifying unknown spatially varying coefficients of a wave equation from measurement data on a lateral boundary. This problem is of interest to many researchers working in various applied fields. For example, we can mention inverse problems related to non-destructive testing techniques and the geophysical problem of finding properties of geophysical media by observations of wave fields on a part of the surface of the earth.

This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, special attention is paid to the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space of an oriented Riemannian four-manifold.

We propose to solve the two-way time domain acoustic wave equation in a generalized Riemannian coordinate system via finite-differences. The coordinate system is defined in such a way that one of its independent variables conforms to the primary wavefront, for example, using a ray coordinate system with the traveltime being one of the coordinates. At each finite-difference time-step, the solution domain is limited to a narrow corridor around the primary wavefront, leading to an increase in the computational performance. A new finite-difference scheme is introduced to stabilize the solution and facilitate its implementation. This new scheme is a blend of the simple explicit and the stable implicit schemes. As a proof of concept, the proposed method is compared to the classical explicit finite-difference scheme performed in Cartesian coordinates on two synthetic velocity models with varying complexities. At a reduced cost, the proposed method produces similar results to the classical one; however, some amplitude differences arise due to various implementation issues. The most direct application for the proposed method is the source side of reverse time migration.