arXiv research

Sharp convergence theorem for Yang-Mills flow on ALE manifolds proved.

The paper proves parabolic gap theorems for Yang-Mills energy.

The paper proves gap results for self-shrinkers in $r$-mean curvature flow.

The paper proves smoothness of transition layers in the Allen-Cahn equation.

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a un…

We use Beltrami's theorem as an excuse to present some arguments from parabolic differential geometry without any of the parabolic machinery.

By adapting methods of \cite{AC} we prove a sharp estimate on the expansion modulus of the gradient of the log of the parabolic kernel to the Schördinger operator with convex potential, which improves an earlier work of Brascamp-Lieb. We also include alternate proofs to the improved log-concavity estimate, and to the f…

The paper proves nonexistence results for certain parabolic inequalities on Riemannian manifolds.

Complete complex parabolic geometries (including projective connections and conformal connections) are flat and homogeneous. This is the first global theorem on parabolic geometries.

Unique submaximal symmetry found for certain parabolic geometries.

Generalized Agol's theorem to 3-manifold groups.

In this paper, we prove the existence and uniqueness theorem for parabolic conical metrics on Riemann surfaces in the situation of generalized real angles, positive, zero and negative, by complex analysis, and give an example of this theorem to clarify concrete expressions of parabolic metrics on the two-sphere and gen…

The paper explores new phenomena in boundaries of relatively hyperbolic groups.

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.

We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and an…

Defines non-parabolic curves in spatial hybrid space with applications.

We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous work by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for (p-)parabolicity. Applications to comparison and uniqueness results involving the p-Laplacian are deduced.

Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.

We prove that any simply connected special Kaehler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. Also we show that a classical result of Calabi and Pogorelov on parabolic spheres implies …

Alternative proof of a theorem using parabolic Monge-Ampère equation in HKT geometry.

The paper defines capacities for minimal graphs over manifolds and proves the half-space property.

The trace set of a Fuchsian group $Γ$ ist the set of length of closed geodesics in the surface $Γ\backslash \mathbb{H}$. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering property. Sarnak then conjectured that the B-C property actually characterizes arithm…

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.

The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.

We prove monotonicity of a parabolic frequency on manifolds. This is a parabolic analog of Almgren's frequency function. Remarkably we get monotonicity on all manifolds and no curvature assumption is needed. When the manifold is Euclidean space and the drift operator is the Ornstein-Uhlenbeck operator this can been see…

The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.

As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in Euclidean spaces. The new feature of our result is the fact that, besides of obtaining a…

In this short note we extend Chow and Lu's advanced maximum principles for parabolic systems on closed manifolds to the case of compact manifolds with boundary, which also generalizes a Hopf type theorem of Pulemotov.

Completes results on complex braid group parabolic subgroups.

Local index theorem for cofinite hyperbolic Riemann surfaces derived from computational perspective.

The paper extends Huber's theorem to higher dimensions using n-Laplace equations.

Proves gap rigidity theorem for Hermitian symmetric spaces.

Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.

Let $\mathbb{M}^{2}$ be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in $\mathbb{M}^{2}\times\mathbb{R}$. First, using a characterization of $δ$-parabolicity we prove that under additional conditions on $\mathbb{M}$,…

A non-elementary Möbius group generated by two-parabolics is determined up to conjugation by one complex parameter and the parameter space has been extensively studied. In this paper, we use the results of \cite{GW} to obtain an additional structure for the parameter space, which we term the {\sl two-parabolic space}. …

In this article we focus on the study of special parabolic points in surfaces arising as graphs of polynomials, we give a theorem of Viro's patchworking type to build families of real polynomials in two variables with a prescribed number of special parabolic points in their graphs. We use this result to build a family …

Improved convergence and curvature estimate for parabolic Allen-Cahn equation.

Researchers derived heat kernel expansions for non-compact spaces using Witten deformation.

The paper extends a theorem about stable minimal surfaces to higher codimensions.

The paper proves gap theorems for Yang-Mills on manifolds with positive Yamabe.

Strong bolicity helps prove Baum-Connes conjecture for certain hyperbolic groups.

In this paper we investigate the moduli space of parabolic Higgs bundles over a punctured Riemann surface with varying weights at the punctures. We show that the harmonic metric depends analytically on the weights and the stable Higgs bundle. This gives a Higgs bundle generalisation of a theorem of McOwen on the existe…

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positiv…

The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.

In this paper, we give an easy proof of the main results of Andrews and Clutterbuck's paper [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916], which gives both a sharp lower bound for the spectral gap of a Schröinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We…

Solves index problem for curved BGG sequences in parabolic geometry.

The paper solves a complex financial optimization problem using a novel mathematical technique.

Combination theorem for PGF groups helps in constructing new examples and understanding their geometry.