arXiv research

Improved gap for mean curvature of biharmonic hypersurfaces in spheres.

The study finds larger gaps in mean curvature for biharmonic submanifolds in spheres.

Paper improves volume gap between minimal submanifolds and unit spheres.

Study spectral gaps in hyperbolic rational homology spheres.

In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely wh…

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

Probabilistic method proves gap estimates on sphere.

Lu conjecture proven for minimal 2-spheres and surfaces under certain conditions.

The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.

Volume gaps for minimal submanifolds in spheres are proven.

Uniform spectral gap found for stable commutator length in hyperbolic 2-orbifolds.

Confirms unique eigenfunction in hyperbolic packing has maximal spectral gap.

Study on Ricci flow on 4-spheres, proving standard sphere convergence.

For an immersed Lagrangian submanifold, let $\check{A}$ be the Lagrangian trace-free second fundamental form. In this note we consider the equation $\nabla^*T=0$ on Lagrangian surfaces immersed in $\mathbb{C}^2$, where $T=-2\nabla^*(\check{A}\lrcornerω)$, and we prove a gap theorem for the Whitney sphere as a solution …

A new screening test for Lasso improves solution speed.

In [SWW], S. Seto, L. Wang and G. Wei proved that the gap between the first two Dirichlet eigenvalues of a convex domain in the unit sphere is at least as large as that for an associated operator on an interval with the same diameter, provided that the domain has the diameter at most $π/2$. In this paper, we extend Set…

Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.

This paper extends gap theorems for submanifolds in hyperbolic space.

We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.

The study proves a gap theorem for CAT(0) spaces with a constant below 1/(6√π).

The paper provides uniform length estimates for trajectories on flat cone surfaces.

Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.

We show that there exists a universal positive constant $\varepsilon_0 > 0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon_0$ where $g_{\mathbb S}$ denot…

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

Special class of surfaces in five-dimensional sphere in $C^3$ is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation $u_{z\bar z}=e^u-e^{-2u}$ which is integrable by means of inverse scattering method. Finite-gap minimal tori are constructed.

Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.

Calibrations help estimate volumes on odd spheres without gaps.

In this paper we consider approximations introduced by Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in [6] about limits of $α$-harmonic maps with uniformly bounded energy. Using a recent energy identity in [7], we obtain an optimal gap theorem for the $α$-harmonic maps…

We show that the family of probability measures on the $n$-dimensional unit sphere, having density proportional to: \[ S^n \ni y \mapsto \frac{1}{|y - x|^{n+α}}, \] satisfies the Curvature-Dimension condition $CD(n-1-\frac{n+α}{4},-α)$, for all $|x| < 1$, $α\geq -n$ and $n\geq 2$. The case $α= 1$ corresponds to the hit…

We consider critical points of the functionals $Π$ and $Ψ$ defined as the global $L^2$-norm of the second fundamental form and mean curvature vector of isometric immersions of compact Riemannian manifolds into a background Riemannian manifold, respectively, as functionals over the space of deformations of the immersion…

Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as the model case. In this article, we extend this result to prove conformally invariant gap theorems for Bach-flat $4$-manifolds with $(\mathbb{CP}^2, g_{FS})$ and $(\mathbb{S}^2\times\mathbb{S}^2,g_{prod…

Flat minimal tori counterexamples refute Lu's second-gap conjecture.

In [SWW16, HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter $D$ of sphere $\mathbb S^n$ is $\geq 3 \frac{π^2}{D^2}$ when $n \geq 3$. We prove the same result when $n=2$. In fact our proof works for all dimension. We also…

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 …

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…

Paper proves a conjecture about minimal hypersurfaces in spheres.

New inequalities for austere submanifolds established.

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

In this sequel to [arXiv:1412.4114], we prove an $L^{d/2}$ energy gap result for Yang-Mills connections on principal $G$-bundles, $P$, over arbitrary, closed, Riemannian, smooth manifolds of dimension $d\geq 2$. We apply our version of the Lojasiewicz-Simon gradient inequality [arXiv:1409.1525, arXiv:1510.03815] to rem…

A new topological gap theorem improves the systole of 3-manifolds with positive scalar curvature.

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the mani…

Extends rigidity results for Whitney spheres in higher dimensions.

We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…

Proves a pinching theorem for self-shrinkers of mean curvature flow.

New bounds on hyperbolic surfaces' properties using linear programming.

A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere $S^3=\R^3\cup \{\infty\}$. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type…

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

The paper proves parabolic gap theorems for Yang-Mills energy.