arXiv research

The paper proves a biharmonic hypersurface in a hemisphere must be a small sphere.

Two embedding methods in spectral graph clustering yield different but valid groupings.

The paper characterizes Pólya's conjecture for spheres and hemispheres, deriving inequalities and bounds.

Computed p-widths for hemisphere, first for manifolds with boundary.

Study proves existence of non-trivial harmonic map flows to hemispheres.

Researchers create metrics on spheres with Ricci curvature ≥1, limiting to Grushin hemisphere.

Study on nonlinear equations on spheres and hemispheres with zero Neumann boundary condition.

The Riemannian hemisphere has a lower bound for its mass.

The hemisphere rigidity theorem connects to the Gelfand problem, providing a precise value for the extremal parameter.

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the …

We study the curvature condition which uniquely characterizes the hemisphere. In particular, we prove the Min-Oo conjecture for hypersurfaces in Euclidean space and hyperbolic space.

Let (M,g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its Yamabe invariant is positive, then (M,g) is conformally isometric to the standard …

Improved upper bound for discrete isometric filling of cycles.

This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. We prove that the only convex billiard without conjugate points on the Hyperbolic plane or on the Hemisphere is circular billiard.

Lower bounds for Dirac eigenvalues on manifolds with boundary.

Paper proves static triples with specific curvature are standard hemispheres.

We prove the following rigidity theorem: For an n-dimensional compact Riemannian manifold with boundary whose Ricci curvature is bounded by n-1 from below, if its boundary is isometric to the standard sphere of dimension n-1 and totally geodesic, then the manifold is isometric to the standard hemisphere.

Extends curves to hemispheres in metric spaces, proving isoperimetric inequalities.

We show that round hemispheres are the only compact 2 dimensional Riemannian manifolds (with or without boundary) such that almost every pair of complete geodesics intersect once and only once. We prove this by establishing a sharp isoperimetric inequality for surfaces with boundary such that every pair of geodesics ha…

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this Gauss map to prove that if $M^{n}$ is a compact, connected and oriented immersed hy…

The paper studies the geometry of eye movements and cycles.

The article characterizes a hemisphere using a Laplace operator and a differential equation.

We characterize the standard $\mathbb{S}^3$ as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: $4π$. As a corollary we answer in the affirmative an interesting special case of a conjecture of Min-Oo's on the scalar curvature rigidity of the upper hemi…

Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature makes $M$ to be isometric to a hemisphere of $\mathbb{S}^{n}$. We also prove that if an Ein…

The paper characterizes D'Atri spaces using total scalar curvature of hemispheres.

Classifies metrics with specific curvature properties on a ball.

The paper proves gap results for constant mean curvature surfaces in Euclidean and hyperbolic spaces.

New rigidity theorems for manifolds with spherical boundary.

Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.

We find that for any n-dimensional, compact, convex subset K of R^{n+1} there is an affinely-spherical hypersurface M in R^{n+1} with center at the relative interior of K, such that the disjoint union of M and K is the boundary of an (n+1)-dimensional, compact, convex set. This so-called affine hemisphere M is uniquely…

New functionals defined for free boundary minimal submanifolds in higher dimensions.

Consider a compact Riemannian manifold M of dimension n whose boundary \partial M is totally geodesic and is isometric to the standard sphere S^{n-1}. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least n(n-1), then M is isometric to the hemisphere S_+^n equipped with its standard metri…

Study on hemisphere threshold for Escobar functional on Riemannian manifolds, revealing mass and boundary invariant behaviors.

Study proves uniqueness of certain minimal surfaces in spherical and hyperbolic spaces.

Paper solves Carathéodory's conjecture for $C^2$-regular convex surfaces.

We retract the scalar curvature rigidity theorem as there is a mistake in the proof. We thank S. Montiel for pointing out the mistake.

Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's conjecture, we prove a volume constrained scalar curvature rigidity theorem which applies to the hemisphere.

Study finds surfaces in spherical caps that maximize modified energy.

The paper studies hypersurfaces with constant weighted mean curvature in Gaussian space.

In this paper, we prove a scalar curvature rigidity result for geodesic balls in S^n. This result contrasts sharply with the recent counterexamples to Min-Oo's conjecture for the hemisphere (cf. [5]).

We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.

New bounds on NTK's smallest eigenvalue for arbitrary data without distributional assumptions.

For all $k,n\ge 1$, we construct a biLipschitz embedding of $\mathbb{S}^n$ into the jet space Carnot group $J^k(\mathbb{R}^n)$ that does not admit a Lipschitz extension to $\mathbb{B}^{n+1}$. Let $f:\mathbb{B}^n\to \mathbb{R}$ be a smooth, positive function with $k^{th}$-order derivatives that are approximately linear …

Deep learning framework detects emotions from EEG data.

The study classifies quasi-Einstein manifolds with boundary.

Extracts causal brain dynamics across multiple scales.

Proves intersection properties of minimal hypersurfaces in various spaces.

The study characterizes Riemannian manifolds with conformal vector fields and proves isometric properties.