Related papers: $C^{\infty} $ spectral rigidity of the ellipse

We prove $C^1$ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients

We prove some rigidity results for compact manifolds with boundary. In particular for a compact Riemannian manifold with nonnegative Ricci curvature and simply connected mean convex boundary, it is shown that if the sectional curvature…

In this paper, we prove a rigidity theorem of asymptotically hyperbolic manifolds only under the assumptions on curvature. Its proof is based on analyzing asymptotic structures of such manifolds at infinity and a volume comparison theorem.

We prove the optimality of the hypotheses guaranteeing the $L^p$-boundedness for the Cauchy-Leray integral in $\mathbb C^n$, $n\geq 2$, obtained in [LS-4]. Two domains, both elementary in nature, show that the geometric requirement of…

In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase…

We study the ellipticity and the ``Nekhoroshev stability'' (stability properties for finite, but very long, time scales) of the Riemann ellipsoids. We provide numerical evidence that the regions of ellipticity of the ellipsoids of types II…

We prove that every compact, pseudoconvex, orientable, CR manifold of $\C^n$, bounds a complex manifold in the $C^\infty$ sense. In particular, the tangential Cauchy-Riemann system has closed range.

Any two compact, complete, one-dimensional geodesic spaces with identical marked length spectrum have isometric $\pi_1$-hull. The present version contains errors, notably in Lemmas 2.2 and 2.3 (path cancellations can be more complicated),…

This paper deals with the lack of compactness in nonlinear elliptic problems $(P)$. In particular, a domain $\Omega$ is provided where not converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that…

We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound $\text{Ric}_{\infty} \geq K > 0$. Our main results show that if the equality holds, the manifold necessarily admits a…

The classical Birkhoff conjecture says that the only integrable convex domains are circles and ellipses. In the paper we show that this a version of this conjecture is true for small perturbations of ellipses of small eccentricity.

The original proof has a gap, and need extra hypothesis that the strong stable and strong unstable filiation both to be $C^1$. The argument is like the following: with the regularity, one can show that the weak-stable and weak-unstable…

We show that every orientable infinite-type surface is properly rigid as a consequence of a more general result. Namely, we prove that if a homotopy equivalence between any two non-compact orientable surfaces is a proper map, then it is…

Suppose that $(M,\mathfrak{g})$ is a compact Riemannian manifold with strictly negative sectional curvatures. A subset of conjugacy classes $E \subset \text{conj}(\pi_1(M))$ is called spectrally rigid if when two negatively curved…

In this paper, we prove a rigidity theorem for Poincar\'e-Einstein manifolds whose conformal infinity is a flat Euclidean space. The proof relies on analyzing the propagation of curvature tensors over the level sets of an adapted boundary…

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that $C^{\infty}$-diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of…

We find some integral formulas of Simons and Bochner type and use them to study biharmonic and biconservative submanifolds in space forms. We obtain rigidity results that in the biharmonic case represent partial answers to two well-known…

The main result of this article is a Llarull-type rigidity statement for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions. The even dimensional analog was proven in an earlier work together with…

A system of quasilinear elliptic equations on an unbounded domain is considered. The existence of a sequence of radially symmetric weak solutions is proved via variational methods.

We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and…