Related papers: The energy spectrum of metrics on surfaces

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked…

In this paper we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be…

We consider some metrics and weak metrics defined on the Teichmueller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and…

We define and study metrics and weak metrics on the Teichmueller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of…

A well-known question asks whether the spectrum of the Laplacian on a Riemannian manifold $(M,g)$ determines the Riemannian metric $g$ up to isometry. A similar question is whether the energy spectrum of all harmonic maps from a given…

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_\rho$ on Teichm\"uller space $\Teich$ which is a qualitative invariant of the holonomy representation…

In this article, we revisit classical length identities enjoyed by simple closed curves on hyperbolic surfaces. We state and prove the rigidity of such identities over Teichm\"uller spaces. Due to this rigidity, certain collections of…

We establish a geometric inequality relating the Dirichlet energy $E_1(f)$ and the bienergy $E_2(f)$ of smooth maps \[ f : (M,g) \to (\overline{M},\overline{g}) \] between Riemannian manifolds. Assume that $(M,g)$ is a compact, connected…

We define a notion of marked length spectrum for $S^1$-symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further…

We consider a closed negatively curved surface $(M, g)$ with marked length spectrum sufficiently close (multiplicatively) to that of a hyperbolic metric $g_0$ on $M$. We show there is a smooth diffeomorphism $F:M \to M$ with derivative…

The rigidity of marked length spectrum for closed hyperbolic surfaces due to Fricke-Klein [7] has been the motivation of many different rigidity results, specially for manifolds of negative curvature. From the works of Vigneras [18], Sunada…

We prove that every closed orientable surface S of negative Euler characteristic admits a pair of finite-degree covers which are length isospectral over S but generically not simple length isospectral over S. To do this, we first…

Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles $>2\pi$. With an additional…

The Steklov spectrum of a smooth compact Riemannian manifold $(M,g)$ with boundary is the set of eigenvalues counted with multiplicities of its Dirichlet-to-Neumann map. (DN map) This article is devoted to the Steklov spectral inverse…

Let $M$ be a closed Riemann surface, $N$ a Riemannian manifold of Hermitian non-positive curvature, $f:M\to N$ a continuous map, and $E$ the function on the Teichm\"uller space of $M$ that assigns to a complex structure on $M$ the energy of…

We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. As a corollary we…

We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed…

A strictly convex real projective orbifold is equipped with a natural Finsler metric called the Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that…

We prove that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the $\epsilon_0$-relative part of Teichmuller spaces of surfaces with boundary.

Given a $C^k$-smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\neq\emptyset$, we consider $\frac 12$-harmonic maps $u\in…