We prove a sharp lower bound for the first nonzero Neumann eigenvalue of geodesic triangles of given diameter in two-dimensional space forms. The bound is given by the first positive radial Neumann eigenvalue of an one dimensional model; it is approached by degenerating isosceles triangles. When \(K>0\) and \(D=\pi/(2\sqrt K)\), equality is attained precisely by birectangular triangles. We also prove a hot-spots theorem for non-acute spherical triangles of diameter at most \(\pi/2\), and establish antisymmetry and eigenvalue monotonicity for isosceles spherical triangles of diameter \(\pi/2\).
We prove a sharp comparison, with Obata-type rigidity, for all Neumann eigenvalues of one-dimensional $\mathrm{CD}(1,2)$ spaces against the Legendre model, under a convexity condition on the density. It follows that minimizers of the $k$-th Laplace eigenvalue among closed surfaces of Gaussian curvature at least $1$ can...
Let $\Omega\subset\mathbb R^n$ be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the $(n-1)$-dimensional volumes of its perpendicular sections. We prove an $O(\varepsilon^2)$ comparison of the mean-zero inverse operato...
We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radi...
We prove the strict log-concavity of the positive first eigenfunction \(-u\) of the \(2\)-Hessian equation and the strict $1/2$-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in $\mathbb{R}^{n}$. As applications, we establish the associated Brunn--Minkowski in...
We establish lower bounds for sums of reciprocals of Laplacian eigenvalues for bounded Euclidean domains, spheres and surfaces. For bounded Euclidean domains, we recover a sharp inequality that extends Bucur and Henrot’s upper bound on the second Neumann eigenvalue. For spheres and surfaces, we obtain new Li- and Yau-t...
Mehdi Eddaoudi· Journal of Spectral Theory· 0 citations
For every integer $d\ge4$, we prove strong stability for a subfamily of classical rotational hypersurfaces in $\mathbb H^d$ with normalized mean curvature one. The examples are complete, two-sided, properly embedded, and nowhere umbilic, with topology $\mathbb {R}\times\mathbb{S}^{d-2}$. An explicit positive supersolut...
Zi-Hao Wang· 1 citation
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