arXiv research

The paper analyzes Karcher means on restricted PSD matrices with statistical guarantees.

New Heintze-Karcher inequality helps understand droplet shapes.

Paper proves inequalities in sub-static warped product manifolds.

Study proves rigidity for Heintze-Karcher inequality in substatic manifolds.

In this article we consider means of positive operators on a Hilbert space. We extend the theory of matrix power means to arbitrary operator means in the sense of Kubo-Ando. The basis of the extension is relying on ideas coming from differential geometry. We consider generalized Karcher equations for positive operators…

Paper proves inequality for capillary hypersurfaces in a wedge.

We propose a conjugate gradient type optimization technique for the computation of the Karcher mean on the set of complex linear subspaces of fixed dimension, modeled by the so-called Grassmannian. The identification of the Grassmannian with Hermitian projection matrices allows an accessible introduction of the geometr…

Proves new inequality for hyperbolic space hypersurfaces.

The paper studies constant mean curvature hypersurfaces in Finsler manifolds.

The Riemannian center of mass was constructed in [GrKa] (1973). In [GKR1, GKR2, Gr, Ka, BuKa] (1974-1981) it was successfully applied with more refined estimates. Probably in 1990 someone renamed it without justification into karcher mean and references to the older papers were omitted by those using the new name. As a…

Paper proves inequality for capillary hypersurfaces with new proof.

Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.

The paper extends Heintze-Karcher inequalities to fractional Q-curvature.

Based on representation theory of Clifford algebra, Ferus, Karcher and Münzner constructed a series of isoparametric foliations. In this paper, we will survey recent studies on isoparametric hypersurfaces of OT-FKM type and investigate related geometric constructions with mean curvature flow.

Karcher reimagined elliptic functions using geometry.

Optimal inequality for free boundary hypersurfaces in convex domains.

In this paper, we prove a generalization of Reilly's formula in \cite{Reilly}. We apply such general Reilly's formula to give alternative proofs of the Alexandrov's Theorem and the Heintze-Karcher inequality in the hemisphere and in the hyperbolic space. Moreover, we use the general Reilly's formula to prove a new Hein…

In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the cost function. The first proposed problem is control for positive definite Hermiti…

Paper proves inequality linking capillary surfaces to Finsler geometry.

The paper shows how to make certain sets on a sphere smooth and flat.

Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.

Improved method for computing Fréchet means on SPD matrices.

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…

In this note we prove the Heintze-Karcher inequality in the context of essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric measure spaces introduced by Cavalletti-Mondino. Mor…

Study flow on de Sitter space for convex hypersurfaces.

In the recent years, Riemannian shape analysis of curves and surfaces has found several applications in medical image analysis. In this paper we present a numerical discretization of second order Sobolev metrics on the space of regular curves in Euclidean space. This class of metrics has several desirable mathematical …

Starting from works by Scherk (1835) and by Enneper-Weierstraß\ (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher (see \cite{Karcher1,Karcher}). In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996 (see \cite{Traizet}).…

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension $n$, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on t…

Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.

This paper extends IMCF theory to Heisenberg group, solving Penrose inequality.

The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.

Stability results for geometric equations in warped product spaces.

Study geometric inequalities for quasi-Einstein manifolds using new formulas.

Study anisotropic capillary surfaces in a wedge using generalized Minkowski norms.

This paper explains a technique for proving geometric inequalities.

The paper proves stability of Wulff shapes using anisotropic curvature functionals.

The paper solves Serrin-type problems on Riemannian manifolds using new inequalities and identities.

Study proves only origin-centered spheres solve certain curvature problems.

In this paper we study the shape space of curves with values in a homogeneous space $M = G/K$, where $G$ is a Lie group and $K$ is a compact Lie subgroup. We generalize the square root velocity framework to obtain a reparametrization invariant metric on the space of curves in $M$. By identifying curves in $M$ with thei…

Study on Serrin's problem in convex cones with rigidity results and geometric inequalities.

Improved Sobolev mappings in Carnot groups with weaker assumptions.

Second order Sobolev metrics on the space of regular unparametrized planar curves have several desirable completeness properties not present in lower order metrics, but numerics are still largely missing. In this paper, we present algorithms to numerically solve the initial and boundary value problems for geodesics. Th…

Using spinorial techniques, we prove, for a class of pseudo-hyperbolic ambient manifolds, a Heintze-Karcher type inequality. We then use this inequality to show an Alexandrov type theorem in such spaces.

Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are non-homogeneous. This generalizes the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.

Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite number of loss functions. The present paper proposes a Riemannian stochastic quasi-Newton algorithm with variance reduction (R-SQN-VR). The key challenges of averaging, adding, and subtracting multipl…

New optimal surfaces found in Heisenberg group defy Euclidean sphere optimality.

Extends isoparametric foliations and area-minimizing cones in product manifolds.

Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is stil…