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We prove that if $(M, g)$ is a compact Riemannian manifold with ergodic geodesic flow, and if $H \subset M$ is a smooth hypersurface satisfying a generic asymmetry condition with respect to the geodesic flow, then restrictions $\phi_j |_H$…
The real homology of a compact, n-dimensional Riemannian manifold M is naturally endowed with the stable norm. The stable norm of a homology class is the minimal Riemannian volume of its representatives. If M is orientable the stable norm…
We study stability properties of $f$-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness…
We prove a local minimizing property for strictly stable free-boundary minimal hypersurfaces in the relative current setting. Let $\Sigma^n$ be a compact, two-sided, properly embedded free-boundary minimal hypersurface in a compact…
Using harmonic mean curvature flow, we establish a sharp Minkowski type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard 3-manifolds. This inequality also improves the known estimates for total…
The Riemannian Penrose inequality is a remarkable geometric inequality between the ADM mass of an asymptotically flat manifold with non-negative scalar curvature and the area of its outermost minimal surface. A version of the Riemannian…
A monotonicity property of Harnack inequality is proved for positive invariant harmonic functions in the unit ball.
We study the differentiability properties of the topological equivalence between a uniformly asymptotically stable linear nonautonomous system and a perturbed system with suitable nonlinearities. For this purpose, we construct a uniformly…
Let $\Omega$ be an open connected cone in $\mathbb{R}^n$ with vertex at the origin. Assume that the operator $$P_\mu:=-\Delta-\frac{\mu}{\delta_\Omega^2(x)}$$ is {\em subcritical} in $\Omega$, where $\delta_\Omega$ is the distance function…
In this work, we show that the completeness relation for the eigenvectors, which is an essential assumption of quantum mechanics, remains true if the Hamiltonian, having a discrete spectrum, is modified by a delta potential (to be made…
In 2015, Mantoulidis and Schoen constructed $3$-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while…
We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic…
We discuss the symmetry factors of Feynman diagrams of scalar field theories with polynomial potential. After giving a concise general formula for them, we present an elementary and direct proof that when computing scattering amplitudes…
We provide integral formulae for the ADM mass of asymptotically flat hypersurfaces in Riemannian manifolds with a certain warped product structure in a neighborhood of infinity, thus extending Lam's recent results on Euclidean graphs to…
We utilize the amenability of the Fermi-type potential profile in Schr{\"o}dinger equation to construct a symmetric one dimensional well as $V(x){=}{-}U_n/[1+\exp[(|x|{-}a)/b]], ~ U_n{=}V_n[1+\exp[-a/b]]$. We define $\alpha=a/b, ~\beta_n…
We prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic $n$-space, $n\geq 3$. The argument uses two new monotone quantities along the inverse mean curvature flow. As an…
We point out an effect which may stabilize a supersymmetric membrane moving on a manifold with boundary, and lead to a light-cone Hamiltonian with a discrete spectrum of eigenvalues. The analysis is carried out explicitly for a closed…
For the PT symmetric potential of Dorey, Dunning and Tateo we show that in the large angular momentum (i.e., strongly spiked) limit the low-lying eigenstates of this popular non-Hermitian problem coincide with the shifted Hermitian harmonic…
In this article, we investigate electrostatic systems with a nonzero cosmological constant on compact manifolds with boundary. We establish new geometric properties for electrostatic manifolds in higher dimensions, extending previous…
By a gradient-like flow on a closed orientable surface $M$, we mean a closed 1-form $\beta$ defined on $M$ punctured at a finite set of points (sources and sinks of $\beta$) such that there exists a Morse function $f$ on $M$, called an…