arXiv research

New expanders found using origami surfaces with spectral gap.

Study proves existence of expanding solutions for multiphase surfaces with regular junctions.

Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.

This paper classifies all expanding Ricci solitons on surfaces.

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

We show that strictly convex surfaces expanding by the inverse Gauss curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.

New CMC existence result for expanding cosmological spacetimes.

The paper examines properties and rigidity of self-expanders in Euclidean space.

Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.

We construct sequences of `expander manifolds' and we use them to show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu. Using expander manifolds in dimension 3 we show that for any $ε, M>0$ there is a Riemannian 3-…

The abstract theorem is extended to higher genus surfaces.

Study on the spectrum of drift Laplacian on Ricci expanders.

Study shows twist tori equidistribute in moduli space, with other families having singular distributions.

Study models Ricci flow on complex surfaces, showing mixed behavior.

In this paper, we first investigate the flow of convex surfaces in the space form $\mathbb{R}^3(κ)~(κ=0,1,-1)$ expanding by $F^{-α}$, where $F$ is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power $α\in(0,1]$ for $κ=0,-1$ and $α=1$ for $κ=1$…

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Min…

Study of Veech surfaces and their twist tori on moduli spaces of abelian differentials.

Streets and Tian introduced a parabolic flow of pluriclosed metrics. We classify the long time behavior of homogeneous solutions of this flow on closed complex surfaces including minimal Hopf, Inoue, Kodaira, and non-Kahler, properly elliptic surfaces. We also construct expanding soliton solutions to the flow on the un…

Study stationary measures and orbit closures for non-abelian actions on surfaces.

Study classifies ruled surfaces from mean curvature flow solutions.

Construct pseudo-Anosovs from expanding interval maps, reconciling Thurston's construction.

Using expander graphs, we construct a sequence of smooth compact surfaces with boundary of perimeter N, and with the first non-zero Steklov eigenvalue uniformly bounded away from zero. This answers a question which was raised in [9]. The genus grows linearly with N, this is the optimal growth rate.

In the present paper we provide new examples of marginally trapped surfaces and tubes in FLRW spacetimes by using a basic relation between these objects and CMC surfaces in 3-manifolds. We also provide a new method to construct marginally trapped surfaces in closed FLRW spacetimes, which is based on the classical Hopf …

Study expanding Ricci solitons on vector bundles, reducing to Higgs bundle equations.

New representations for surface groups expand known Anosov classes.

Research shows surfaces close to planes in Hausdorff distance.

The main objective of this thesis is the study of the evolution under the Ricci flow of surfaces with singularities of cone type. A second objective, emerged from the techniques we use, is the study of families of Ricci flow solitons in dimension 2 and 3. The Ricci flow is an evolution equation for Riemannian manifolds…

This paper extends danceability concept to twisted virtual knots.

Study shows how charged MOTS restrict spacetime configurations.

Due to a significant error in the main result (pointed out by J. Wahl), the paper has been withdrawn by the authors. A corrected and expanded version is 'Rational blow-downs and smoothings of surface singularities' by A. Stipsicz, Z. Szabo and J. Wahl.

Complex analysis aids in studying minimal surfaces.

We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangian…

We consider globally hyperbolic spacetimes with compact Cauchy surfaces in a setting compatible with the presence of a positive cosmological constant. More specifically, for 3+1 dimensional spacetimes which satisfy the null energy condition and contain a future expanding compact Cauchy surface, we establish a precise c…

Extends Heegaard Floer theory to surfaces of dimension one.

In this sequel paper we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient co…

We show that the boundaries of thin strongly pseudoconvex Grauert tubes, with respect to the Guillemin-Stenzel Kähler metric canonically associated with the Poincaré metric on closed hyperbolic real-analytic surfaces, has nowhere vanishing Cartan CR-curvature. This result provides a wealth of examples of compact $3$-di…

We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called "time flat surfaces." S…

This article is an expanded version of the talk given by Ch. O. at the Second Latin Congress on "Symmetries in Geometry and Physics" in Curitiba, Brazil in December 2010. In this version we explain the topological and gauge-theoretical aspects of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of K…

In this paper we survey some recent contributions by the authors to the theory of null holomorphic curves in the complex Euclidean space $\mathbb C^3$, as well as their applications to null holomorphic curves in the special linear group $SL_2(\mathbb C)$, minimal surfaces in the Euclidean space $\mathbb R^3$, and const…

Constructs flow lines connecting unstable to stable self-expanders.

The paper extends rigidity results for $λ$-self-expanders to hyperplanes, spheres, and cylinders.

The study explores dilating set properties across Euclidean and hyperbolic geometries.

New self-expander found between two given asymptotic ones.

No expanding breathers found in noncompact Ricci flows with certain curvature conditions.

As part of the general investigation of Ricci flow on complete surfaces with finite total curvature, we study this flow for surfaces with asymptotically conical (which includes as a special case asymptotically Euclidean) geometries. After establishing long-time existence, and in particular the fact that the flow preser…

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the 3-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is determined by the initial surface. To prove…

Solves Ricci flow on Riemann surfaces with measure initial data.

New expanders for mean curvature flow contradict genus-reduction conjecture.