We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if $M$ is compact, is defined for $w \geq 0$ and $T > 0$ by $$N_{M,w}(T) = \sum\limits_{λ_n \leq T}(T-λ_n)^w $$ where $\{λ_n\}$ is the set of eigenvalues of the Laplacian which acts on the space of smooth functions on $M$. If $M$ is non-compact, then the weighted counting function is defined via the inverse Laplace transform. Now let $M_{\ell}$ denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface $M_{0}$. As an example of our results, we have the following theorem: There is an explicitly defined function $G_{\ell,w}(T)$ which depends solely on $\ell$, $w$, and $T$ such that for $w > 3/2$ and $T>0$, we have $$N_{M_{\ell},w}(T) = G_{\ell,w}(T) +N_{M_{0},w}(T) +o(1)$$ for $\ell \to 0$. We also consider the setting when $w < 3/2$, and we obtain a new proof of the continuity of small eigenvalues on degenerating hyperbolic Riemann surfaces of finite volume.
We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars and horizontal boundary collars, we establish a generalized energy identity.
We prove a gluing theorem for solutions of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface representing a boundary point of Teichmüller moduli space.
We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity. We find conditions that are both necessary and sufficient for the compactness in $W^{1,2}$ and $C^{0}$ modulo bubbles of sequences of such maps.
In this paper we prove a uniform estimate for the gradient of the Green function on a closed Riemann surface, independent of its conformal class, and we derive compactness results for immersions with L2-bounded second fundamental form and for riemannian surfaces of uniformly bounded gaussian curvature entropy.
Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by the Narasimhan-Simha measure. For $m = 1$, both these measures coincide with the Bergman meas…
We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial surface has some symmetries. If these symmetries are not present we show that the first eigenvalue normalized by area strictly decreases for the same range of parameters. These results are the main motivation for the construction in \cite{MS3}, where we show a monotonicity result f…
Given a hyperbolic surface and a simple closed geodesic on it, complex-twists along the curve produce a holomorphic family of deformations in Teichmüller space, degenerating to the Riemann surface where it is pinched. We show there is a corresponding Teichmüller disk such that the two are strongly asymptotic, in the Teichmüller metric, around the noded Riemann surface. We establish a similar comparison with plumbing deformations that open the node.
We consider harmonic diffeomorphisms to a fixed hyperbolic target $Y$, from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray, to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on $Y$. The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of …
In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an $n$-dimensional Riemannian manifold.
The aim of this paper is to associate a measure for certain sets of paths in the Euclidean plane $\mathbb{R}^2$ with fixed starting and ending points. Then, working on parameterized surfaces with a specific Riemannian metric, we define and calculate the integral of the length over the set of paths obtained as the image of the initial paths in $\mathbb{R}^2$ under the given parameterization. Moreover, we prove that this integral is given by the average of the lengths of the external paths times the measure of the set of paths if and only if the surface has Gaussian curvature equal to zero.
Let $(M, {g})$ be a compact, $d$-dimensional Riemannian manifold without boundary. Suppose further that $(M,g)$ is either two dimensional and has no conjugate points or $(M,g)$ has non-positive sectional curvature. The goal of this note is to show that the long time parametrix obtained for such manifolds by Bérard can be used to prove a logarithmic improvement for the remainder term of the Riesz means of the counting function of the Laplace operator.
We study the behaviors of the relative Bergman kernel metrics on holomorphic families of degenerating hyperelliptic Riemann surfaces and their Jacobian varieties. Near a node or cusp, we obtain precise asymptotic formulas with explicit coefficients. In general the Bergman kernels on a given cuspidal family do not always converge to that on the regular part of the limiting surface, which is different from the nodal case. It turns out that information on both the singularity and complex structure contributes to various asymptotic behaviors of the Bergman kernel. Our method involves the classical…
Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the resolvent trace coefficients, the residues of the spectral zeta function as well as certain Wodzicki residues. We then use the Wodzicki residue to obtain results about the local counting function of operators of Dirac and Laplace type. In particular, we express the second term of the…
On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate for $d+δ$ and $Δ_q$. Then we obtain a complete characterization of the range of $p>1$ for which the Riesz transform for $q$-forms $T_q=(d+δ)Δ_q^{-1/2}$ is bounded on $L^p$. Finally, we obtain an asymptotic formula for the analytic torsion of a family of smooth compact Riemannian…
Let $Ω$ be a bounded domain with $C^\infty$ boundary in an $n$-dimensional $C^\infty$ Riemannian manifold, and let $\varrho$ be a non-negative bounded function defined on $\partial Ω$. It is well-known that for the biharmonic equation $Δ^2 u=0$ in $Ω$ with the 0-Dirichlet boundary condition, there exists an infinite set $\{u_k\}$ of biharmonic functions in $Ω$ with positive eigenvalues $\{λ_k\}$ satisfying $Δu_k+ λ_k \varrho \frac{\partial u_k}{\partial ν}=0$ on the boundary $\partial Ω$. In this paper, we give the Weyl-type asymptotic formula with a sharp remainder estimate for the counting f…
Let M be a complete Riemannian manifold with negative curvature, and let C_-, C_+ be two properly immersed closed convex subsets of M. We survey the asymptotic behaviour of the number of common perpendiculars of length at most s from C_- to C_+, giving error terms and counting with weights, starting from the work of Huber, Herrmann, Margulis and ending with the works of the authors. We describe the relationship with counting problems in circle packings of Kontorovich, Oh, Shah. We survey the tools used to obtain the precise asymptotics (Bowen-Margulis and Gibbs measures, skinning measures). We…
