Non-compact Surfaces Research Articles

Sharp Decay Estimates for the Logarithmic Fast Diffusion Equation and the Ricci Flow on Surfaces

We prove the sharp local L^1-L^infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p-L^infty estimate for p>1. It also has several applications in geometry, providing the missing step in order to pose the Ricci flow with rough initial data in the noncompact case, for example starting with a general noncompact Alexandrov surface, and giving the sharp asymptotics for the contracting cusp Ricci flow, as we show elsewhere.

Veech groups of infinite-genus surfaces

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $\mathbf{GL}_+(2,\R)$ avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surface is either countable, or one of three specific types.

Quantum Bound States on Some Hyperbolic Surfaces

The aim of this work is to highlight results of energy eigenstates on some noncompact finite hyperbolic surfaces. Such systems are known to exhibit both continuous and discrete spectra and are dependent on the subgroups of the modular group that underlie these surfaces. We study explicitly the cases of Maass cusp forms on the singly punctured two-torus and the triply punctured two-sphere for their eigenvalues. The eigenvalues for the torus system are doubly degenerate while for the sphere case, the eigenvalues are nondegenerate. We also note that the lowest eigenvalue of the sphere system is larger than that of the torus system.

PROPER AFFINE DEFORMATIONS OF THE ONE-HOLED TORUS

A Margulis spacetime is a complete at Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface ∑ homotopy-equivalent to M. The purpose of this paper is to classify Margulis spacetimes when ∑ is homeomorphic to a one-holed torus. We show that every such M decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of ∑. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces.

On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form

In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern–Osserman inequality. In the particular case of a surface of nonnegative curvature, we prove that the surface is diffeomorphic to the Euclidean plane if the surface has tamed second fundamental form, and that the surface is isometric to the Euclidean plane if the surface has strongly tamed second fundamental form. In the last part of the paper we characterize the fundamental tone of any submanifold of tamed second fundamental form immersed in an ambient space with a pole and quadratic decay of the radial sectional curvatures.

Martin compactification of a complete surface with negative curvature

In this paper we consider the Martin compactification, associated with the operator $$\mathcal {L}= \Delta -1$$ , of a complete non-compact surface $$\Sigma ^2$$ with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator $$\Delta $$ of $$\Sigma ^2$$ , and prove a uniqueness result: such eigenfunctions are unique up to a positive multiplicative constant, if they vanish on the part of the geometric boundary $$S_\infty (\Sigma ^2)$$ of $$\Sigma ^2$$ where the curvature is bounded above by a negative constant, and satisfy some growth estimate on the other part of $$S_\infty (\Sigma ^2)$$ where the curvature approaches zero. This uniqueness result plays an essential role in our recent paper (Cao and He, Infinitesimal rigidity of steady gradient Ricci soliton in dimension three. arXiv:1412.2714 , 2014, preprint), in which we prove an infinitesimal rigidity theorem for deformations of certain three-dimensional collapsed gradient steady Ricci soliton with a non-trivial Killing vector field.

The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces

We construct distinctive surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we prove that a complete noncompact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary if the manifold M is not less curved than a noncompact model surface of revolution and if the total curvature of the model surface is finite and less than 2π. By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.

On the Prime Geodesic Theorem for Non-Compact Riemann Surfaces

We use B. Randol’s method to improve the error term in the prime geodesic theorem for a noncompact Riemann surface having at least one cusp. The case considered is a general one, corresponding to a Fuchsian group of the first kind and a multiplier system with a weight on it.

Spectral Determinants on Mandelstam Diagrams

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric $${|\omega|^2}$$ , where $${\omega}$$ is a meromorphic one-form with simple poles such that all its periods are pure imaginary and all its residues are real. The main result is an explicit formula for the determinant of the Laplacian in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic form $${\omega}$$ . As an important intermediate result we prove a decomposition formula of the type of Burghelea–Friedlander–Kappeler for the determinant of the Laplacian for flat surfaces with cylindrical ends and conical singularities.

On the wave representation of hyperbolic, elliptic, and parabolic Eisenstein series

We develop a unified approach to the construction of the hyperbolic and elliptic Eisenstein series on a finite volume hyperbolic Riemann surface. Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein series as integral transforms of the kernel of a wave operator. Established results in the literature relate the wave kernel to the heat kernel, which admits explicit construction from various points of view. Therefore, we obtain a sequence of integral transforms which begins with the heat kernel, obtains a Poisson and wave kernel, and then yields the hyperbolic and elliptic Eisenstein series. In the case of a non-compact finite volume hyperbolic Riemann surface, we finally show how to express the parabolic Eisenstein series in terms of the integral transform of a wave kernel.

Common zeroes of families of smooth vector fields on surfaces

Let Y and X denote $$C^k$$ vector fields on a possibly noncompact surface with empty boundary, $$1\le k <\infty $$ . Say that Y tracks X if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of zeroes of X. Theorem Assume the Poincare–Hopf index of X at K is nonzero, and the k-jet of X at each point of K is nontrivial. If $$\mathfrak {g}$$ is a supersolvable Lie algebra of $$C^k$$ vector fields that track X, then the elements of $$\mathfrak {g}$$ have a common zero in K. Applications are made to the dynamics of attractors and transformation groups.

Antiperfection of Yang–Mills Morse theory over a nonorientable surface

We use Morse theory of the Yang–Mills functional to compute the Betti numbers of the moduli stack of flat U(3)-bundles over a compact nonorientable surface. Our result establishes the antiperfection conjecture of Ho–Liu, and establishes the equivariant formality conjecture of the author for U(3)-bundles.

A Poincaré–Bendixson theorem for meromorphic connections on Riemann surfaces

We shall prove a Poincare–Bendixson theorem describing the asymptotic behavior of geodesics for a meromorphic connection on a compact Riemann surface. We shall also briefly discuss the case of non-compact Riemann surfaces, and study in detail the geodesics for a holomorphic connection on a complex torus.

Universality in mean curvature flow neckpinches

We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is $C^3$-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singular set, and will have a unique tangent flow.

Construction of hyperbolic Riemann surfaces with large systoles

Let S be a compact hyperbolic Riemann surface of genus \({g \geq 2}\). We call a systole a shortest simple closed geodesic in S and denote by \({{\rm sys}(S)}\) its length. Let \({{\rm msys}(g)}\) be the maximal value that \({{\rm sys}(\cdot)}\) can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface Smax a compact Riemann surface of genus g whose systole has length \({{\rm msys}(g)}\). In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating \({{\rm msys}(\cdot)}\) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.

Remarks on the Einstein–Euler-Entropy system

We prove short-time existence for the Einstein–Euler-Entropy system for non-isentropic fluids with data in uniformly local Sobolev spaces. The cases of compact as well as non-compact Cauchy surfaces are covered. The method employed uses a Lagrangian description of the fluid flow which is based on techniques developed by Friedrich, hence providing a completely different proof of earlier results of Choquet-Bruhat and Lichnerowicz. This new proof is specially suited for applications to self-gravitating fluid bodies. Along the way, we review some basic definitions and ideas, giving thus a relatively self-contained exposition that also serves as an introduction to many aspects of the problem.

Causal and conformal structures of two-dimensional globally hyperbolic spacetimes

The group of conformal diffeomorphisms and the group of causal automorphisms on two-dimensional globally hyperbolic spacetimes are clarified. It is shown that if spacetimes have non-compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Minkowski spacetime, and if spacetimes have compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Einstein's static universe.

Strichartz Inequalities on Surfaces with Cusps

We prove Strichartz inequalities for the wave and Schrodinger equations on noncompact surfaces with ends of finite area, i.e. with ends isometric to (r0, ∞) × S 1 , dr 2 + e −2φ(r) dθ 2 with e −φ integrable. We prove first that all Strichartz estimates, with any derivative loss, fail to be true in such ends. We next show for the wave equation that, by projecting off the zero mode of S 1 , we recover the same inequalities as on R 2. On the other hand, for the Schrodinger equation, we prove that even by projecting off the zero angular modes we have to consider additional losses of derivatives compared to the case of closed surfaces; in particular, we show that the semiclassical estimates of Burq-Gerard-Tzvetkov do not hold in such geometries. Moreover our semiclassical estimates with loss are sharp.

Vector-valued Automorphic Forms and Vector Bundles

While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group $\Gamma$ and an arbitrary representation $R$ of $\Gamma$ in GL$(n,{\mathbb C})$, their existence has been established in the literature only when restrictions are imposed on both $\Gamma$ and $R$. In this paper, we prove the existence of $n$ linearly independent vector-valued automorphic forms for any Fuchsian group $\Gamma$ and any $n$-dimensional complex representation $R$ of $\Gamma$. To this end, we realize these automorphic forms as global sections of a special rank $n$ vector bundle built using solutions to the Riemann-Hilbert problem over various noncompact Riemann surfaces and Kodaira's vanishing theorem.

On finite index subgroups of the mapping class group of a nonorientable surface

Let $M(N_{h,n})$ denote the mapping class group of a compact nonorientable surface of genus $h\ge 7$ and $n\le 1$ boundary components, and let $T(N_{h,n})$ be the subgroup of $M(N_{h,n})$ generated by all Dehn twists. It is known that $T(N_{h,n})$ is the unique subgroup of $M(N_{h,n})$ of index $2$. We prove that $T(N_{h,n})$ (and also $M(N_{h,n})$) contains a unique subgroup of index $2^{g-1}(2^g-1)$ up to conjugation, and a unique subgroup of index $2^{g-1}(2^g+1)$ up to conjugation, where $g=\lfloor(h-1)/2\rfloor$. The other proper subgroups of $T(N_{h,n})$ and $M(N_{h,n})$ have index greater than $2^{g-1}(2^g+1)$. In particular, the minimum index of a proper subgroup of $T(N_{h,n})$ is $2^{g-1}(2^g-1)$.