Related papers: Generalized boundary rigidity and minimal surface …
A recent analysis of real general relativity based on multisymplectic techniques has shown that boundary terms may occur in the constraint equations, unless some boundary conditions are imposed. This paper studies the corresponding form of…
This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis…
We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.
In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts.…
We consider an $\mathcal{MP}$-system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple $\mathcal{MP}$-systems, we study the $\mathcal{MP}$-ray transform in order to obtain new…
A Finsler metric is geodesically reversible if geodesics remain geodesics after a change of orientation. Asymmetric norms on vector spaces and Funk metrics in the interior of convex bodies are examples of geodesically reversible metrics…
In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalar flat metric having constant mean curvature. This result can be seen as a generalization…
Uniform bounds are developed for derivatives of solutions of the $2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichm\"{u}ller and moduli spaces. The dependence of the bounds on the geometry of…
We prove that non-trivial bounds for generalized Radon transforms imply correspondingly non-trivial discrete incidence theorems for manifolds and suitably regular point sets.
We show that the real-analytic matrix-weighted double fibration transform determines the analytic wavefront set of a vector-valued function. We apply this result to show that the matrix weighted ray transform is injective on a…
The concept of sharpness has been successfully applied to traditional architectures like MLPs and CNNs to predict their generalization. For transformers, however, recent work reported weak correlation between flatness and generalization. We…
We are concerned with the relaxation and existence theories of a general class of geometrical minimisation problems, with action integrals defined via differential forms over fibre bundles. We find natural algebraic and analytic conditions…
Geodesic balls in a simply connected space forms $\mathbb{S}^n$, $\mathbb{R}^{n}$ or $\mathbb{H}^{n}$ are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible…
We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator…
The purpose of this paper is to prove the L^p boundedness of singular Radon transforms and their maximal analogues. These operators differ from the traditional singular integrals and maximal functions in that their definition at any point x…
One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e…
Optimization on Riemannian manifolds widely arises in eigenvalue computation, density functional theory, Bose-Einstein condensates, low rank nearest correlation, image registration, and signal processing, etc. We propose an adaptive…
In this work, we study compact Riemannian manifolds with boundary satisfying V-static-type equations. By combining a generalized Reilly formula with Steklov-type boundary value problems, we derive integral inequalities for geometric…
We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and…
We study metric spaces homeomorphic to a closed oriented manifold from both geometric and analytic perspectives. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary,…