arXiv research

The paper explores gaps in curvature-related metrics and rigidity.

Proves gap rigidity theorem for Hermitian symmetric spaces.

Study rigidity of spectral gap on Finsler manifolds with specific curvature bounds.

New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.

Study proves rigidity and gap theorems for specific metrics.

We consider a rigidity problem for the spectral gap of the Laplacian on an $RCD(K,\infty)$-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive $K$. For a weighted Riemannian manifold, Cheng--Zhou showed that the sharp spectral gap is achieved only when a $1$-dimensional G…

Paper proves Simon's third gap conjecture for minimal surfaces in spheres.

In this paper, we study the properties of the first global term in the polyhomogeneous expansions for Liouville's equation. We obtain rigidity and gap results for the boundary integral of the global coefficient. We prove that such a boundary integral is always nonpositive, and is zero if and only if the underlying doma…

Paper studies a new curvature system and proves rigidity and gap theorems.

Survey of rigidity and gap phenomena in sphere-ball submanifolds.

Since $n$-dimensional $λ$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $λ$-hypersurfaces. We give a gap theorem of complete $λ$-hypersurfaces with po…

Local gaps in Ricci shrinkers depend only on dimension.

Study on Kähler manifolds shows rigidity of eigenvalues with positive Ricci bound.

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian case correspond to one-sided bounds on the sectional curvatures. Starting from 2-d…

We extend Obata's rigidity theorem to free probability.

In our previous work "Characterization of certain homorphic geodesic cycles on Hermitian locally symmetric manifolds of the noncompact type" in "Modern methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we formulated a conjecture: the so called "gap phenomenon". The purpose of the article is two-fo…

We study $λ$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_Σ e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete $λ$-hypersurfaces in terms of the norm of the second fundamental form $|A|$. Sec…

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

Develops correlation number for specific potentials and Hitchin representations.

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap …

The paper proves various inequalities on gradient shrinking Ricci solitons.

High-dimensional curved diffusions show abrupt convergence at a critical time.

Optimizes dimension estimate for holomorphic functions on Kähler manifolds.

Let f:(Y,g)->(X,g_0) be a non zero degree continuous map between compact Kähler manifolds of dimension greater or equal to 2, where g_0 has constant negative holomorphic sectional curvature. Adapting the Besson-Courtois-Gallot barycentre map techniques to the Kähler setting, we prove a gap theorem in terms of the degre…

We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of projective measured laminations that preserve such a stratification is in general identif…

The paper proves rigidity results for certain supergravity backgrounds in 11 dimensions.

In this paper, we study compact generalized $τ$-quasi Ricci-harmonic metrics. In the first part, we explore conditions under which generalized $τ$-quasi Ricci-harmonic metrics are harmonic-Einstein and give some characterization results for it. In the second part, we obtain some rigidity results for compact $(τ, ρ)$-qu…

Volume gaps for minimal submanifolds in spheres are proven.

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor,…

Extends rigidity results for Whitney spheres in higher dimensions.

It is proved by Brendle in [4] that the equatorial disk $D^k$ has least area among $k$-dimensional free boundary minimal surfaces in the Euclidean ball $B^n$. By comparing the excess of free boundary minimal surfaces with the excess of the associated cones over the boundary, we prove the existence of a gap for the area…

The paper proves a rigidity theorem for minimal submanifolds in spheres with flat normal bundle.

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{…

New $\varepsilon$-harmonic maps of low degree are rigid under certain energy bounds.

ASAM improves deep neural network generalization by adapting sharpness to scale.

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature est…

In this paper we show that the only properly immersed self--shrinkers $Σ$ in $\mathbb{R}^{m+1}$ with Morse index $1$ are the hyperplanes through the origin. Moreover, we prove that if $Σ$ is not a hyperplane through the origin then the index jumps and it is at least $m+2$, with equality if and only if $Σ$ is a cylinder…

The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy…

The study proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.

The paper proves a gap theorem for almost non-negatively curved manifolds.

The paper studies special Lagrangian submanifolds in complex spaces and derives inequalities and flow methods.

Study on deformation of affine structures on Lie groups using cohomology.

A classical question in spectral geometry is, for each pair of nonnegative integers $(p,n)$ such that $p\leq 2n$, if the eigenvalues of Laplacian on $p$-forms of a compact Kähler manifold are the same as those of $\mathbb{C}P^n$ equipped with the Fubini-Study metric, then whether or not this Kähler manifold is holomorp…

Calibrations help estimate volumes on odd spheres without gaps.

The paper proves manifold rigidity for specific scalar curvature conditions.

We prove that complete warped product Einstein metrics with isometric bases, simply connected space form fibers, and the same Ricci curvature and dimension are isometric. In the compact case we also prove that the warping functions must be the same up to scaling, while in the non-compact case there are simple examples …

A contact stationary Legendrian submanifold (briefly, CSL submanifold) is a stationary point of the volume functional of Legendrian submanifolds in a Sasakian manifold. Much effort has been paid in the last two decades to construct examples of such manifolds, mainly by geometers using various geometric methods. But we …