Search

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such submanifolds. These inequalities provide estimations of the mean curvature (the main extrinsic invariants) in terms of intrinsic invariants: Ricci curvature, the Chen invariant, and scalar curvature. In the proofs, we use the sectional curvature of a semi-symmetric, non-metric connection recently defined by A. Mihai and the first author, as well as its properties. [ABSTRACT FROM AUTHOR]

We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the "singular Yamabe problem". [ABSTRACT FROM AUTHOR]

In this paper, we establish a generalized Wintgen inequality for quaternionic bi-slant submanifolds and QR-submanifolds (with minimal codimension) in quaternion space forms. We also aim to characterize the second fundamental form of those submanifolds for which the equality cases can hold. Finally, we provide examples of submanifolds embedded in quaternion space forms to support our results. [ABSTRACT FROM AUTHOR]

We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local nature. It can thus be applied to any surface in a region not containing characteristic points. We provide a geometrical interpretation of the coefficients appearing in the expansion, and compute them on some relevant examples in three-dimensional sub-Riemannian model spaces. These results generalize those obtained in [Z. M. Balogh, F. Ferrari, B. Franchi, E. Vecchi and K. Wildrick, Steiner's formula in the Heisenberg group, Nonlinear Anal. 126 (2015) 201–217; M. Ritoré, Tubular neighborhoods in the sub-Riemannian Heisenberg groups, Adv. Calc. Var. 14(1) (2021) 1–36] for the Heisenberg group. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study rigidity of free boundary minimal two-disks that locally maximize the modified Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature and mean convex boundary. Assuming the strict stability of Σ , we prove that a neighborhood of it in M is isometric to one of the half de Sitter–Schwarzschild space. [ABSTRACT FROM AUTHOR]

We study a ramification of a phenomenon discovered by Baird and Eells (in: Looijenga et al (eds) Geometry Symposium Utrecht 1980. Lecture Notes in Mathematics, Springer, Berlin, 1981) i.e. that non-constant harmonic morphisms Φ : M N → N 2 from a N -dimensional ( N ≥ 3 ) Riemannian manifold M N , into a Riemann surface N 2 , can be characterized as those horizontally weakly conformal maps having minimal fibres. We recover Baird–Eells’ result for S 1 invariant harmonic morphisms Φ : M 2 n + 2 → N 2 from a class of Lorentzian manifolds arising as total spaces M = C (M) of canonical circle bundles S 1 → M → M over strictly pseudoconvex CR manifolds M 2 n + 1 . The corresponding base maps ϕ : M 2 n + 1 → N 2 are shown to satisfy lim ϵ → 0 + π H ϕ μ ϵ V ϕ = 0 , where μ ϵ V ϕ is the mean curvature vector of the vertical distribution V ϕ = Ker (d ϕ) on the Riemannian manifold (M , g ϵ) , and { g ϵ } 0 < ϵ < 1 is a family of contractions of the Levi form of the pseudohermitian manifold (M , θ) . [ABSTRACT FROM AUTHOR]

After Thomas James Willmore, many authors were looking for an immersion of a manifold in Euclidean space or Riemannian manifold, which is the critical point of functionals whose integrands depend on the mean curvature or the norm of the second fundamental form. We study a new Willmore-type variational problem for a foliated hypersurface in Euclidean space. Its general version is the Reilly-type functional, where the integrand depends on elementary symmetric functions of the eigenvalues of the restriction on the leaves of the second fundamental form. We find the 1st and 2nd variations of such functionals and show the conformal invariance of some of them. For a critical hypersurface with a transversally harmonic foliation, we derive the Euler-Lagrange equation and give examples with low-dimensional foliations. We present critical hypersurfaces of revolution and show that they are local minima for special variations of immersion.

The Wintgen inequality is a significant result in the field of differential geometry, specifically related to the study of submanifolds in Riemannian manifolds. It was discovered by Pierre Wintgen. In the present work, we deal with the Riemannian maps between Riemannian manifolds that serve as a superb method for comparing the geometric structures of the source and target manifolds. This article is the first to explore a well-known conjecture, called DDVV inequality (a conjecture for Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken), for Riemannian maps, where we consider different space forms as target manifolds. There are numerous research problems related to such inequality in various ambient manifolds. These problems can all be explored within the general framework of Riemannian maps between various Riemannian manifolds equipped with notable geometric structures. [ABSTRACT FROM AUTHOR]

A surface is minimal if the mean curvature ℋ mean {\mathcal{H}_{\rm mean}} vanishes everywhere. In this paper, we study some surfaces in the product space ℍ 2 × ℝ {\mathbb{H}^{2}\times\mathbb{R}} . In particular, we completely classify minimal surfaces. [ABSTRACT FROM AUTHOR]

Applying a family of mass-capacity related inequalities proved in Miao (Peking Math J 2023, ), we obtain sufficient conditions that imply the nonnegativity as well as positive lower bounds of the mass, on a class of manifolds with nonnegative scalar curvature, with or without a singularity. [ABSTRACT FROM AUTHOR]

On the two-sphere Σ , we consider the problem of minimising among suitable immersions f : Σ → R 3 the weighted L ∞ norm of the mean curvature H, with weighting given by a prescribed ambient function ξ , subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of "pseudo-minimiser" surfaces must satisfy a second-order PDE system obtained as the limit as p → ∞ of the Euler–Lagrange equations for the approximating L p problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: H ∈ { ± ‖ ξ H ‖ L ∞ } away from the nodal set of the PDE system, and H = 0 on the nodal set (if it is non-empty). [ABSTRACT FROM AUTHOR]

After Thomas James Willmore, many authors were looking for an immersion of a manifold in Euclidean space or Riemannian manifold, which is the critical point of functionals whose integrands depend on the mean curvature or the norm of the second fundamental form. We study a new Willmore-type variational problem for a foliated hypersurface in Euclidean space. Its general version is the Reilly-type functional, where the integrand depends on elementary symmetric functions of the eigenvalues of the restriction on the leaves of the second fundamental form. We find the 1st and 2nd variations of such functionals and show the conformal invariance of some of them. For a critical hypersurface with a transversally harmonic foliation, we derive the Euler-Lagrange equation and give examples with low-dimensional foliations. We present critical hypersurfaces of revolution and show that they are local minima for special variations of immersion. [ABSTRACT FROM AUTHOR]

The generalized Zermelo navigation problem looks for the shortest time paths in an environment, modeled by a Finsler manifold (M, F), under the influence of wind or current, represented by a vector field W. The main objective of this paper is to investigate the relationship between the isoparametric functions on the manifold M with and without the presence of the vector field W. Our work generalizes results in (Dong and He in Differ Geom Appl 68:101581, 2020; He et al. in Acta Math Sinica Engl Ser 36:1049–1060, 2020; He et al. in Differ Geom Appl 84:101937, 2022; Ming et al. in Pub Math Debr 97:449–474, 2020; Xu et al. in Isoparametric hypersurfaces induced by navigation in Lorentz Finsler geometry, 2021). For the positive-definite cases, we also compare the mean curvatures in the manifold. Overall, we follow a coordinate-free approach. [ABSTRACT FROM AUTHOR]

This paper is devoted to introducing and investigating the bounded variation capacity and the perimeter in the abstract Wiener space X, thereby discovering some related inequalities. Functions of bounded variation in an abstract Wiener space X have been studied by many scholars. As the continuation of this research, we define the corresponding BV capacity cap H ⁡ (⋅) (now called abstract Wiener BV capacity) and investigate its properties. We also investigate some properties of sets of finite γ-perimeter, with γ being a Gaussian measure. Subsequently, the isocapacitary inequality associated with cap H ⁡ (⋅) is presented and we are able to show that it is equivalent to the Gaussian isoperimetric inequality. Finally, we prove that every set of finite γ-perimeter in X has mean curvature in L 1 ⁢ (X , γ) . [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose an Eulerian type of numerical finite difference approximation on structured grids for an advection dominated level set equation enhanced with a small curvature term. The proposed numerical scheme is based on upwind discretization in both the advection and the curvature term. This class of approximations is used in algebraic solvers, such as fast sweeping or fast marching methods, which respect the causality of solutions for which any information is propagating along characteristics, in order to solve the algebraic system in a finite number of computationally simple iterations or updates. An implicit numerical scheme for a time relaxed level set equation with non-zero right-hand side is used in this paper and solved by a nonlinear fast sweeping method. The results of the von Neumann stability analysis of the proposed scheme are presented, and numerical examples are used to verify the analysis. [ABSTRACT FROM AUTHOR]

Willmore defined embedded surfaces on f : S → E 3 , which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of E 3 , inducing a Riemannian structure on f (S) . The expression analogous to the left-hand member of the curvature K is replaced by the mean curvature H 2 on f (S). Our aim is to observe the Gaussian and mean curvatures of curve–surface pairs using embedded surfaces in different curve–surface pairs and to define some developable operations on their curve–surface pairs. We also investigate the embedded surfaces using the Willmore method. We first recall the Darboux curve–surface and derive the new characterizations. This curve–surface pair is called the osculating Darboux curve–surface if its position vector always lies in the osculating Darboux plane spanned by a Darboux frame. Thus, we observed an osculating Darboux curve–surface pair. We also obtained the D ∼ -scroll of the curve–surface pair and involute D ∼ -scroll of the curve–surface pair with some differential geometric elements and found D ∼ (α , M) (s) and D ∗ ∼ (α , M) (s) -scrolls of the curve–surface pair (α , M) . [ABSTRACT FROM AUTHOR]

We prove a general result characterizing a specific class of Serrin domains as supports of unbounded and periodic constant mean curvature graphs. We apply this result to prove the existence of a family of unbounded periodic constant mean curvature graphs, each supported by a Serrin domain and intersecting its boundary orthogonally, up to a translation. We also show that the underlying Serrin domains are calibrable and Cheeger in a suitable sense, and they solve the 1-Laplacian equation. [ABSTRACT FROM AUTHOR]

Deep brain stimulation (DBS) is an effective treatment for neurological disorders, and accurately reconstructing the DBS lead trajectories is crucial for MRI compatibility assessment and surgical planning. This paper presents a novel fully automated framework for reconstructing DBS lead trajectories from postoperative CT images. The leads were first segmented by thresholding, but would be fused together somewhere. Mean curvature analysis of multi-layer CT number isosurfaces was introduced to effectively address lead fusion, due to the different topological characteristics of the isosurfaces in and out of the fusion regions. The position of electrode contacts was determined through morphological analysis to get the starting point and the initial direction for trajectory tracking. The next trajectory point was derived by calculating the weighted average coordinates of the candidate points, using the distance from the current estimated trajectory and the CT number as weights. This method has demonstrated high accuracy and efficiency, successfully and automatically reconstructing complex bilateral trajectories for 13 patient cases in less than 10 minutes with errors less than 1 mm. This work overcomes the limitations of existing semi-automatic techniques that require extensive manual intervention. It paves the way for optimizing DBS lead trajectory to reduce tissue heating and image artifacts, which will contribute to neuroimaging studies and improve clinical outcomes. Code for our proposed algorithm is publicly available on Github.

Mean curvature is fundamentally important for various research fields, including differential geometry in mathematics, cell membrane modeling in biophysics and image processing in computer vision. However, its optimization is a challenging task. Conventionally, its optimization can be carried out by Partial Differential Equations (mean curvature flows) or solved by Augmented Lagrangian Methods. These approaches are numerically slow because they have to satisfy several numerical stability constraints. To tackle this issue, in this paper, we propose to use deep neural networks for the mean curvature optimization. Our method has several advantages. First, the proposed network has only 0.046M parameters, leading to a compact model. Second, the proposed network can be trained on low resolution images but applied on any resolution images. Third, it is flexible in the loss function design and thus can be used as a generic optimization method. To numerically valid the effectiveness and efficiency of the proposed method, several experiments are conducted. The results confirm that the proposed method achieves lower energy than traditional methods and thus establishes a new the-state-of-the-art for mean curvature optimization. The proposed method can be applied in a large range applications, where mean curvature optimization is involved.

In this article, we investigate the Kenmotsu manifold when applied to a \(D_{\alpha}\)-homothetic deformation. Then, given a submanifold in a \(D_{\alpha}\)-homothetically deformed Kenmotsu manifold, we derive the generalized Wintgen inequality. Additionally, we find this inequality for submanifolds such as slant, invariant, and anti-invariant in the same ambient space.

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such submanifolds. These inequalities provide estimations of the mean curvature (the main extrinsic invariants) in terms of intrinsic invariants: Ricci curvature, the Chen invariant, and scalar curvature. In the proofs, we use the sectional curvature of a semi-symmetric, non-metric connection recently defined by A. Mihai and the first author, as well as its properties.

In this paper, we study 4-dimensional complete hypersurfaces with w-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for 4-dimensional complete hypersurfaces with w-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei [Adv. Math. 314 (2017), pp. 278–305] under the weaker topological condition. [ABSTRACT FROM AUTHOR]

By J.F. Nash's Theorem, any Riemannian manifold can be embedded into a Euclidean ambient space with dimension sufficiently large. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash's Theorem effectively is finding useful relationships between intrinsic and extrinsic elements that are characterizing immersions. After 1993, when a groundbreaking work written by B.-Y. Chen on this theme was published, many explorations pursued this important avenue of inquiry. Bearing in mind this historical context, in our present paper we point out a new relationship involving intrinsic and extrinsic curvature invariants, under natural geometric conditions. We conclude our work with an obstruction to minimality, in the spirit of S.-S. Chern's reflection from 1968. [ABSTRACT FROM AUTHOR]

In classical differential geometry, the problem of obtaining Gaussian and mean curvatures of a surface is one of the most important problems. A surface M² in I³ is a THA-surface of first type if it can be parameterized by r(s, t) = (s, t, Af(s + at)g(t) + B(f(s + at) + g(t))). A surface M² in I³ is a THA-surface of second type if it can be parameterized by r(s, t) = (s, Af(s + at)g(t) + B(f(s + at) + g(t)), t), where A and B are non-zero real numbers [16, 17, 18]. In this paper, we classify two types THA-surfaces in the 3-dimensional isotropic space I³ and study THA-surfaces with zero curvature in I³. [ABSTRACT FROM AUTHOR]

We prove explicit upper and lower bounds for the Poisson hierarchy, the averaged L1-moment spectra A k B R M vol S R M k = 1 ∞ , and the torsional rigidity A 1 (B R M) of a geodesic ball B R M in a Riemannian manifold Mn which satisfies that the mean curvatures of the geodesic spheres S r M included in it, (up to the boundary S R M ), are controlled by the radial mean curvature of the geodesic spheres S r ω (o ω) with same radius centered at the center oω of a rotationally symmetric model space M ω n . As a consecuence, we prove a first Dirichlet eigenvalue λ 1 (B R M) comparison theorem and show that equality with the bound λ 1 (B R ω (o ω)) , (where B r ω (o ω) is the geodesic r-ball in M ω n ), characterizes the L1-moment spectrum A k (B R M) k = 1 ∞ as the sequence A k (B R ω) k = 1 ∞ and vice-versa. [ABSTRACT FROM AUTHOR]

In this paper Gaussian, Mean and total curvature formulae of Rational Bezier Curves with asymptotic frame field are calculated by using its curvatures in 3- dimensional lightlike cone in Minkowski 4- Space. Our main intention is to introduce and investigate some differential geometric properties of the the Rational Bezier Curves with asymptotic frame field in 3- dimensional lightlike cone in Minkowski 4- Space by using its curvatures. [ABSTRACT FROM AUTHOR]

Bump hunting deals with finding in sample spaces meaningful data subsets known as bumps. These have traditionally been conceived as modal or concave regions in the graph of the underlying density function. We define an abstract bump construct based on curvature functionals of the probability density. Then, we explore several alternative characterizations involving derivatives up to second order. In particular, a suitable implementation of Good and Gaskins' original concave bumps is proposed in the multivariate case. Moreover, we bring to exploratory data analysis concepts like the mean curvature and the Laplacian that have produced good results in applied domains. Our methodology addresses the approximation of the curvature functional with a plug-in kernel density estimator. We provide theoretical results that assure the asymptotic consistency of bump boundaries in the Hausdorff distance with affordable convergence rates. We also present asymptotically valid and consistent confidence regions bounding curvature bumps. The theory is illustrated through several use cases in sports analytics with datasets from the NBA, MLB and NFL. We conclude that the different curvature instances effectively combine to generate insightful visualizations. [ABSTRACT FROM AUTHOR]

A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.

In this article, we study the uniqueness of positive symmetric solutions of the following mean curvature problem in Euclidean space: (P)u′1+∣u′∣2′+h(x)f(u)=0,−1

In this paper, we study the conchodial surfaces in 3-dimensional Euclidean space with the condition $\Delta x_{i}=\lambda _{i}x_{i}$ where $\Delta $ denotes the Laplace operator with respect to the first fundamental form. We obtain the classification theorem for these surfaces satisfying under this condition. Furthermore, we have given some special cases for the classification theorem by giving the radius function $r(u,v)$ with respect to the parameters $u$ and $v$.

In this paper, we will develop a different technique to study the rigidity of complete $ \lambda $-translators $ x:M^{2} \rightarrow \mathbb R^{3} $ with the non-zero constant gauss curvature in the Euclidean space $ \mathbb R^{3} $.

In this paper, we obtain a stochastic criterion for a minimal surface in terms of transition densities of stochastic processes generated by the fundamental forms of the surface. [ABSTRACT FROM AUTHOR]

The twisted hypersurfaces x with the (0 , 0 , 0 , 0 , 1) rotating axis in five-dimensional Euclidean space E 5 is considered. The fundamental forms, the Gauss map, and the shape operator of x are calculated. In E 5 , describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces x are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of x are determined. Additionally, the Laplace–Beltrami operator relation of x is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of conformally flat spaces that includes the space forms as well as the doubled Schwarzschild space, and, we will prove an extension of Jellett's theorem. Next, we build an example, where the extension of Jellett's theorem does not hold. We also prove a Heintze–Karcher‐type inequality and we presented a one‐parametric family of conformally flat spaces, all distinct from space forms, where it holds a Heintze–Karcher‐type inequality. [ABSTRACT FROM AUTHOR]

In this study, we first define the Smarandache curves derived from the Frenet vectors and the Darboux vector of any curve. Then, we construct new ruled surfaces along these Smarandache curves with the direction vectors obtained from the Frenet vectors and the Darboux vector, and give the equations of these surfaces. In addition, we calculate the Gaussian and mean curvatures of these surfaces separately and present the conditions to be minimal and developable for these surfaces. Finally, as an example, we obtain ruled surfaces whose base curves are Viviani's curves and plot the graphics of these surfaces. [ABSTRACT FROM AUTHOR]

We consider a classical (capillary) model for a one-phase liquid in equilibrium. The liquid (e.g., water) is subject to a volume constraint, it does not mix with the surrounding vapour (e.g., air), it may come into contact with solid supports (e.g., a container), and it is subject to the action of an analytic potential field (e.g., gravity). The region occupied by the liquid is described as a set of locally finite perimeter (Caccioppoli set) in R³; no a priori regularity assumption is made on its boundary. The (twofold) scope in this note is to propose a weakest possible set of mathematical assumptions that sensibly describe a condition of stable equilibrium for the liquid-vapour interface (the capillary surface), and to infer from those that this interface is a smoothly embedded analytic surface. (The liquid-solid-vapour junction, or free boundary, can be present but is not analysed here.) The result relies fundamentally on the recent varifold regularity theory developed by the author and Wickramasekera, and on the identification of a suitable formulation of the stability condition. [ABSTRACT FROM AUTHOR]

A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the conchodial surfaces in 3-dimensional Euclidean space with the condition ∆xi = λixi where ∆ denotes the Laplace operator with respect to the first fundamental form. We obtain the classification theorem for these surfaces satisfying under this condition. Furthermore, we have give some special cases for the classification theorem by given the radius function r(u, v) with respect to the parameter u and v. [ABSTRACT FROM AUTHOR]

If both the arc length and the intrinsic curvature of a curve or surface are preserved, then the flow of the curve or surface is said to be inextensible. The absence of motion-induced strain energy is the physical characteristic of inextensible curve and surface flows. In this paper, we study inextensible tangential, normal and binormal ruled surfaces generated by a curve with constant torsion, which is also called a Salkowski curve. We investigate whether or not these surfaces are minimal or can be developed. In addition, we prove some theorems which are related to inextensible ruled surfaces within three-dimensional Euclidean space.

Willmore defined embedded surfaces on f:S→E3, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of E3, inducing a Riemannian structure on f(S). The expression analogous to the left-hand member of the curvature K is replaced by the mean curvature H2 on f(S). Our aim is to observe the Gaussian and mean curvatures of curve–surface pairs using embedded surfaces in different curve–surface pairs and to define some developable operations on their curve–surface pairs. We also investigate the embedded surfaces using the Willmore method. We first recall the Darboux curve–surface and derive the new characterizations. This curve–surface pair is called the osculating Darboux curve–surface if its position vector always lies in the osculating Darboux plane spanned by a Darboux frame. Thus, we observed an osculating Darboux curve–surface pair. We also obtained the D∼-scroll of the curve–surface pair and involute D∼-scroll of the curve–surface pair with some differential geometric elements and found D∼(α,M)(s) and D∗∼(α,M)(s)-scrolls of the curve–surface pair (α,M).

We introduce the notion of harmonic curvature for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2$Q^m = SO_{m+2}/SO_mSO_2$. First, we prove that the unit normal vector field N on real hypersurfaces in Qm$Q^m$ with harmonic curvature is singular, that is, A$\mathfrak {A}$‐principal or A$\mathfrak {A}$‐isotropic. Next by using a new notion of cyclic semiparallel Ricci tensor, we give a new result on real hypersurfaces with harmonic curvature and constant mean curvature in Qm=SOm+2/SOmSO2$Q^m = SO_{m+2}/SO_mSO_2$. [ABSTRACT FROM AUTHOR]