arXiv research

Simple Ricci flow proof for Riemann surfaces.

Uniform proof for Ricci flows on complete manifolds.

We prove effective uniformization for nearly round 2-spheres and investigate their stability.

New proof of uniformization for hyperbolic foliations.

Uniform proof of property R_infinity for specific Artin-Tits groups.

We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterizati…

Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.

The paper studies invariant weighted Bergman metrics on domains.

In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow…

In this note we clarify that the Rcci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.

New proof of Kähler-Einstein Fano manifold $L^\infty$ estimates.

Uniform proof for ultradifferentiability in various classes and dimensions.

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant…

Simplified proof for asset pricing theory.

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The $C^{k,α}$-smoothness result is o…

We show that the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto, while introducing new arguments using homology to certify that certain curves are nonsepara…

Proof shows volumes of certain geometric representations are always integers.

The paper proves a conjecture about manifold limits and characterizes their structure.

Uniform proof reconstructs spaces using cross ratio on boundary.

We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact Kähler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of the uniform estimate does not rely on the exsitence of Kähler-Einstein metrics on $M…

Starting with a model conical Kähler metric, we prove a uniform scalar curvature bound for solutions to the conical Kähler-Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also establish uniform estimates for the potentials and their time derivatives.

Simplified proof and new $C^0$ estimate for Kähler-Einstein metrics.

Proof of boundedness of quasimorphisms for certain Lie groups.

We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization the…

The note provides uniform estimates for complex Hessian equations on compact Hermitian manifolds.

Uniform hyperbolicity proved for nonorientable surface curve graphs.

Uniform bounds for Green's function on Kähler manifolds derived from complex Monge-Ampère equations.

The uniform boundary condition in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the $\ell^1$-norm on the singular chain complex, Matsumoto and Morita established a characterisation of the uniform boundary condition in terms of bounded cohomology. In particular, spaces…

In recent work, we have proven uniform decay bounds for solutions of the wave equation $\Box_gφ=0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $|φ|\le Cv_+^{-1}$, which holds throughout the domain of outer communications, where $v$ is an advanced Eddington-Finkelstein coordinate, $v_+=\ma…

The study of the geometry of $n$-uniform measures in $\mathbb{R}^{d}$ has been an important question in many fields of analysis since Preiss' seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is on…

Uniform estimates for elliptic problems near polygonal domains.

This paper has been withdrawn by the author due to an error in an inequality in the proof of Theorem 1.1.

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any suc…

We prove that the curve graph $\calC^{(1)}(S)$ is Gromov-hyperbolic with a constant of hyperbolicity independent of the surface $S$. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.

We give a maximum principle proof of interior derivative estimates for the Kähler-Ricci flow, assuming local uniform bounds on the metric.

Equivalence proven between uniformizing varieties and tensors, generalizing uniformization results.

Length metrics can be closely approximated by conformally flat metrics.

We give a proof that there exists a universal constant $K$ such that the disc graph associated to a surface $S$ forming a boundary component of a compact, orientable 3-manifold $M$ is $K$-quasiconvex in the curve graph of $S$. Our proof does not require the use of train tracks.

We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's $κ$ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some ear…

Adapting \cite{strz3}, we define generalized $p$-harmonic maps into Riemannian homogeneous targets, a notion of solutions not belonging to the energy space. Restricting our attention to the subcritical range $p$ greater than the domain dimension $n$, we show a uniform $C^{1,α}$-regularity result for a sequence of such …

We provide a constructive, variational proof of Rivin's realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corres…

Geodesic rays prove key aspects of cscK metrics existence and stability.

We present two proofs of the fact, originally due to Reiner Martin, that any fully irreducible hyperbolic element of $Out(F_N)$ acts on the projectivized space of geodesic currents $\mathbb{P}Curr(F_N)$ with uniform north-south dynamics. The first proof, using purely train-track methods, provides an elaborated and corr…

Uniform RC-positivity results for direct image bundles.

Study improves understanding of why agentic theorem provers succeed.

Uniformly proves index invariance for signature operators on manifolds.

By using the De Giorgi iteration method we will give a new simple proof of the recent result of B.Kotschwar, O.Munteanu, J.Wang [KMW] and N.Sesum [S] on the local boundedness of the Riemmanian curvature tensor of solutions of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci…

Study Neumann problem for special Lagrangian type equations.