arXiv research

Classifies $π_1$-injective maps between non-compact surfaces.

Strong rigidity proven for non-compact surfaces.

Goldman bracket distinguishes surface homeomorphisms.

Paper calculates homotopy types of non-compact surfaces using groupoids.

Maps between non-compact surfaces can have geometric kernels under certain conditions.

Bonahon's method for compactifying Teichmüller space extended to non-compact surfaces.

This paper studies non-compact Ricci surfaces with catenoidal ends.

Complete solutions found for Toda equations on non-compact surfaces.

Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.

Study shows unique harmonic metrics for certain Higgs bundles over non-compact surfaces.

In this note, we prove that the holonomy map from the set of equivalence classes of projective structures of parabolic type on non compact surfaces to the set of equivalence classes of parabolic representations of the fundamental group of the surface to P SL 2 (C) is a local biholomorphism.

We study non-compact surfaces obtained by gluing strips $\mathbb{R}\times(-1,1)$ with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the gro…

A Heegaard splitting of an open 3-manifold is the partition of the manifold into two non-compact handlebodies which intersect on their common boundary. This paper proves several non-compact analogues of theorems about compact Heegaard splittings. The main theorem is: if N is a compact, connected, orientable 3-manifold …

Harmonic metrics are established for SO0(n,n)-Higgs bundles on non-compact hyperbolic surfaces.

We will construct 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 will prove that a complete non-compact Riemannian manifold M is homeomorphic to t…

We prove that for any non-compact connected surface $M$ the group $H_c(M)$ of compactly suported homeomorphisms of $M$ endowed with the Whitney topology is homeomorphic to $R^\infty\times l_2$ or $Z\times R^\infty\times l_2$.

The study finds surface subgroups in cocompact lattices of $H^{2n}$ for $n\geq2$.

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 surf…

Researchers prove mirror symmetry for certain non-compact Calabi-Yau surfaces.

This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.

The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface $S_{m,s}$ with $s$ ends and exactly $m$ of them with infinite genus, such that $m,s\in \mathbb{N}$ and $1<m\leq s$, we give a precise description of t…

For appropriately values of $H$, we obtain an area estimate for a complete non-compact $H$-surface of finite topology and finite area, embedded in a three-manifold of negative curvature. Moreover, in the case of equality and under additional assumptions, we prove that a neighbourhood of the mean convex side of the surf…

Let $Z$ be a non-compact two-dimensional manifold and $Δ$ be a one-dimensional foliation of $Z$ such that $\partial Z$ consists of leaves of $Δ$ and each leaf of $Δ$ is a non-compact closed subset of $Z$. We obtain a characterization of a subclass of such foliated surfaces $(Z,Δ)$ glued from open strips $\mathbb{R}\tim…

Harmonic metrics on Higgs bundles over non-compact hyperbolic surfaces are studied.

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ ``of finite uniton type", from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type. This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces. To be mo…

We show the existence of several new families of non-compact constant mean curvature surfaces: (i) singly-punctured surfaces of arbitrary genus $g \geq 1$, (ii) doubly-punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.

The paper studies spaces of non-compact real algebraic curves and their uniformisation.

We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m,1) (m>0) contain surface subgroups.

Unified rigidity theorem for cyclic and alternating surfaces.

The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.

Paper develops a new method for harmonic maps into symmetric spaces.

We give a number of examples of pairs of non-compact surfaces which are isoscattering, and which are exceptionally simple in one or more senses. We give examples which are of small genus with a small number of ends, and also examles which are congruence surfaces.

New minimal surface theory disproves a conjecture in symmetric spaces.

We study Hawking mass and the Huisken's isoperimetric mass evaluated on surfaces with boundary. The convergence to an ADM mass defined on asymptotically flat manifold with a non-compact boundary are proved.

In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following…

On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a metric of constant curvature and, for negative Euler characteristic, exhibited a flow from a given metric to a constant cu…

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = Γ\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $Γ$ containing parabolic elements. $\mathcal M$ is t…

We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space $\real^{2,1}$. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group $SU_2$ with $SU_{1,1}$. The non…

In this paper we prove that if the geodesic flow of a {compact or non-compact} complete manifold without conjugate points is of the Anosov type, then the average of the integral of the sectional curvature along the geodesic is negative and away from zero from a uniform time. Moreover, in dimension two, if the manifold …

Study on Dirac operator spectrum on shrinking surfaces with cusps.

The paper extends the collar theorem to non-compact surfaces using new comparison theorems.

The paper studies how certain surfaces evolve in space-time.

Non-compact convex sets in hyperbolic 3-space are rigid under isometries.

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\…

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This dete…

We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semisimple Lie groups (e.g. SL(n,C)/SU(n)), which contains minimal surfaces in R^n and constant mean curvature 1 surfaces in H^3. A Weierstrass type representation formula, and a Chern-Osserman type inequality for s…

It was shown by Ramanathan \cite{R} that any compact oriented non-simply-connected minimal surface in the three-dimensional round sphere admits at most a finite set of pairwise noncongruent minimal isometric immersions. Here we show that this result extends to isotropic surfaces in spheres of arbitrary dimension. The c…

Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projecti…