arXiv research

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…

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

Classifies metrics with specific curvature properties on a ball.

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

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

Improved upper bound for discrete isometric filling of cycles.

In Euclidean and Hyperbolic space, and the hemisphere in $S^n$, geodesic balls maximize the gap $λ_2 - λ_1$ of Dirichlet eigenvalues, amoung domains with fixed $λ_1$. We prove an upper bound on $λ_2 - λ_1$ for domains in manifolds with certain curvature bounds. The inequality is sharp on geodesic balls in spaceforms.

We study conformal deformation problems on manifolds with boundary which include prescribing $σ_k\equiv0$ in the interior. In particular, we prove a Dirichlet principle when the induced metric on the boundary is fixed and an Obata-type theorem on the upper hemisphere. We introduce some conformally covariant multilinear…

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

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

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 …

New characterizations for manifolds with boundary rigidity results.

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 identifies obstructions for solving a 4th-order boundary problem.

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.

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.

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 …

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-Émery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m…

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 …

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.

Rigidity theorem for special metrics on 4-manifolds.

Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2γ\in(0,2)$ or $2γ\in(2,4)$ and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption…

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.

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…

In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with the zero Neumann boundary condition, which are so-called Brezis-Nirenberg type problems, and we give conditions on which such equations have only constant solutions. We also extend these results for…

The paper studies the geometry of eye movements and cycles.

The study proves unique static manifolds with positive scalar curvature and boundary.

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

Let $(M,g)$ be a compact manifold with boundary and $Ric_g\geq (n-1)g$, Hang and Wang proved that $(M,g)$ is isometric to the standard hemisphere if $\partial M$ is convex and isometric to $\mathbb{S}^{n-1}(1)$. We prove some rigidity theorems when $\partial M$ is isometric to a product manifold where one factor is th…

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.

In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $Σ$ is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then $Σ$ is either …

Let $X$ be a Banach space or more generally a complete metric space admitting a conical geodesic bicombing. We prove that every closed $L$-Lipschitz curve $γ:S^1\rightarrow X$ may be extended to an $L$-Lipschitz map defined on the hemisphere $f:H^2\rightarrow X$. This implies that $X$ satisfies a quadratic isoperimetri…

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…

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.

In this paper, we prove that the static triple with half harmonic Weyl curvature and positive scalar curvature must be the standard hemisphere.

Study finds surfaces in spherical caps that maximize modified energy.

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

Clustering is concerned with coherently grouping observations without any explicit concept of true groupings. Spectral graph clustering - clustering the vertices of a graph based on their spectral embedding - is commonly approached via K-means (or, more generally, Gaussian mixture model) clustering composed with either…

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.