arXiv research

Every noncompact surface has a 3-rigid triangulation.

Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.

Study mapping class groups of infinite type surfaces with noncompact boundaries.

Study finds conditions for minimal surfaces in noncompact spaces.

The paper extends topological field theory to noncompact surfaces using symmetric powers.

Avoids noncompact hypersurfaces from touching in evolving flows.

Study end sum for surfaces and prove uniqueness results.

Complex equivalence classes found in graph homotopy.

Study combinatorial Yamabe flow on infinite triangulated surfaces.

The paper constructs noncompact hyperbolic surfaces with uniform spectral gaps using random graph models.

In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. We prove that a complete noncompact Kähler surface with positive and bounded sectional curvature and with finite analytic Chern number $c_{1}(M)^{2}$ is biholomorphic to ${\C}^2$.

The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.

Minimal surfaces span periodic curves in 3D space.

The study of potential functions on noncompact quasi-Einstein manifolds, focusing on dimensions and flatness.

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 larger than 1. It also has several applications in geometry, providing …

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed $3$-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family…

Existence proof of noncompact self-shrinkers with arbitrary genus.

We explain how the current knowledge on the set of complete noncompact constant mean curvature surfaces can be exploited to produce new examples of compact constant mean curvature surfaces of genus greater than or equal to 3.

Minimal hyperbolic foliations on 3-manifolds have non-simply connected generic leaves.

This paper contains a correction of a mistake made in arXiv:1405.1324

Classifies surfaces for pure mapping class groups with automatic continuity.

Let $(M,g)$ be a globally symmetric space of noncompact type, of arbitrary rank, and $Δ$ its Laplacian. We prove the existence of a meromorphic continuation of the resolvent $(Δ-\ev)^{-1}$ across the continuous spectrum to a Riemann surface multiply covering the plane. The methods are purely analytic and are adapted fr…

No CMC surfaces exist in certain hyperbolic 3-manifolds.

We prove that a complete noncompact Kähler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}$^{n}$ and we show that the manifold is topologically {\bf R}$^{2n}$. In particular, when $M^{n}$ is a Kähler surface of positive bisecti…

The paper classifies Toda equations for noncompact symmetric spaces and their solutions.

In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold $M$ of positive and bounded holomorphic bisectional curvature, suppose its…

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).

We investigate projective properties of Lorentzian surfaces. In particular, we prove that if T is a non flat torus, then the index of its isometry group in its projective group is at most two. We also prove that any topologically finite noncompact surface can be endowed with a metric having a non isometric projective t…

Type II (ancient) solutions to the Ricci flow on surfaces are not yet classified. It is conjectured that the Rosenau solution and the cigar are the only solutions, modulo scaling. In this paper, we mainly study the backward limit and the circumference at spatial infinity of Type II ancient solutions on noncompact surfa…

We give a short proof of the following fact. Let $Σ$ be a connected, finitely connected, noncompact manifold without boundary. If $g$ is a complete Riemannian metric on $Σ$ whose Gaussian curvature $K$ is nonnegative at infinity, then $K$ must be integrable. In particular, we obtain a new short proof of the fact that i…

The paper proves a special case of Yau's conjecture for Kähler surfaces.

Study covers of surfaces, showing types and properties.

We obtain a Chern-Osserman type equality of a complete properly immersed surface in Euclidean space, provided the L^2-norm of the second fundamental form is finite. Also, by using a monotonicity formula, we prove that if the L^2-norm of mean curvature of a noncompact surface is finite, then it has at least quadratic ar…

The paper confirms a conjecture for 3D manifolds and extends it to 3-7D under specific conditions.

Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deform…

We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry group the natural $\bfZ_2$ extension of the dihedral group $D_n$. The surfaces are …

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich's estimate for manifolds with positive scalar curvature as well as the author's estimate on surfaces.

We extend the unpublished work of M. Handel and R. Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel-Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic lamaniations, show the geodesic laminations satisfy the axioms, and p…

Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane w…

Built on a recent work of Almaraz, Barbosa, de Lima on positive mass theorems on asymptotically flat manifods with a noncompact boundary, we apply free boundary minimal surface techniques to prove their positive mass theorem and study the existence of positive scalar curvature metrics with mean convex boundary on a con…

Given a surface of higher genus, we will look at the Weil-Petersson completion of the Teichmuller space of the surface, and will study the isometric action of the mapping class group on it. The main observation is that the geometric characteristics of the setting bear strong similarities to the ones in semi-simple Lie …

Estimates submanifold diameters in curved spaces.

For each k > 0 we find an explicit function f_k such that the topology of S inside the ball B(p,r) is `bounded' by f_k(r) for every complete Riemannian surface (compact or noncompact) with K\geq -k^2, every point p on the surface, and every r. Using this result, we obtain a characterization (simple to check in practica…

It is shown that the mass of an asymptotically flat manifold with a noncompact boundary can be computed in terms of limiting surface integrals involving the Einstein tensor of the interior metric and the Newton tensor attached to the second fundamental form of the boundary. This extends to this setting previous results…

In this work, we obtain some existence results of Chern-Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as $t\rightarrow 0$. These results can be viewed as generalization of an existence result by Giesen and Topping f…

In this paper we study some global properties of static potentials on asymptotically flat $3$-manifolds $(M,g)$ in the nonvacuum setting. Heuristically, a static potential $f$ represents the (signed) length along $M$ of an irrotational timelike Killing vector field, which can degenerate on surfaces corresponding to the…

Extends Llarull's theorem to noncompact manifolds with boundary.

The study proves uniqueness of large isoperimetric sets in specific noncompact manifolds.