The study introduces metrics on parabolic bundles and proves their positivity properties.
problem Positivity properties of metrics on parabolic bundles.
method Introduced admissible Hermitian metrics, developed Chern-Weil theory, and formulated a Griffiths conjecture.
result Griffiths conjecture holds for certain parabolic structures on Riemann surfaces.
The study of special metrics in various parabolic geometries.
problem Finding special metrics in parabolic geometries.
method Investigating invariant conditions for special metrics in different parabolic geometries.
result Characterization of special metrics in hypersurface CR and contact Legendrean cases.
The paper proves existence and uniqueness of parabolic metrics on Riemann surfaces with conical singularities.
problem Existence and uniqueness of parabolic metrics on Riemann surfaces with conical singularities.
method Complex analysis
result Proves existence and uniqueness of parabolic metrics on Riemann surfaces with conical singularities.
Reproves results on spherical metrics using parabolic bundles.
problem Existence and uniqueness of conformal spherical metrics with prescribed angles.
method Kobayashi-Hitchin correspondence for parabolic bundles.
result Reproves Troyanov and Luo-Tian's results.
Sol3 is shown not to be parabolic.
problem Proving non-parabolicity of Sol3 method Analyzing left invariant metrics on Sol3 result Sol3 is non-parabolic Gradient estimates for nonlinear parabolic equations on metric spaces.
problem Nonexistence of positive solutions to nonlinear parabolic equations.
method Local elliptic and parabolic gradient estimates.
result Conditions guaranteeing the nonexistence of nontrivial positive solutions.
Extends parabolic study to flat hyperkähler manifolds.
problem Study problems in hyperhermitian geometry.
method Extends elliptic approach to parabolic setting.
result Solves problems in hyperhermitian geometry.
Nonexistence results for semilinear parabolic and hyperbolic inequalities on metric graphs
problem Nonexistence of solutions to semilinear parabolic and hyperbolic inequalities on metric graphs
method Construction of a new pseudo-metric and space-time test functions
result All solutions must be identically zero
The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
method Using Souplet-Zhang type estimates and properties of Bakry-Emery Ricci tensor and weighted mean curvature.
result Gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with Dirichlet boundary condition.
New examples of extremal Kähler metrics on blow-ups of parabolic ruled surfaces are constructed. The method is based on the gluing construction of Arezzo, Pacard and Singer. This enables to endow ruled surfaces of the form P(O⊕L) with special parabolic structures such that the associated iter…
The paper defines a path metric on a stable component of polynomial families.
problem Understanding the geometry of polynomial families with parabolic relations.
method Constructing a positive semi-definite pressure form on a bounded stable component of the moduli space.
result The pressure form defines a path metric on the stable component.
Study proves correspondence for special bundles on complex surfaces.
problem Proving correspondence for parabolic bundles on complex surfaces.
method Kobayashi-Hitchin correspondence for parabolic bundles over compact non-Kähler surfaces.
result Proved Kobayashi-Hitchin correspondence for specified bundles.
New method for creating metrics in complex geometries.
problem Metrizability of parabolic geometries and sub-Riemannian metrics.
method General method for linearizability and classification of cases.
result Natural sub-Riemannian metrics on distributions in geometric control theory.
Solves a parabolic equation on manifolds for Gauduchon metrics.
problem Proves existence and uniqueness of solution to a parabolic Monge-Ampère equation.
method Long time existence and uniqueness proof using normalization convergence.
result Normalization of solution converges to a smooth function as time goes to infinity.
Study on Higgs bundles and hyperpolygon spaces using Hitchin metrics.
problem Investigating the Hitchin metric on moduli spaces of Higgs bundles.
method Using Hitchin hyperkähler metric and parabolic Deligne-Hitchin moduli space.
result Rescaled Hitchin metric converges to hyperpolygon space's hyperkähler metric in the semiclassical limit.
New flow deforms Riemannian metrics smoothly.
problem Deforming Riemannian metrics on spin manifolds.
method Parabolic flow based on Dirac-Einstein functional.
result Local well-posedness of smooth solutions proved.
Analyzes harmonic metrics for Higgs bundles over punctured surfaces.
problem Analyzing the moduli space of parabolic Higgs bundles with varying weights.
method Investigates the analytic dependence of harmonic metrics on weights and stable Higgs bundles.
result Shows the harmonic metric depends analytically on weights and stable Higgs bundles.
In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line b…
Study on conditions for Randers metrics to be of constant Ricci curvature.
problem Conditions for Randers metrics to be of constant Ricci curvature.
method Analysis of sufficient and necessary conditions for Randers metrics with and without strong convexity.
result Classification of Randers metrics with ∥β∥α>1 and ∥β∥α≡1. Survey on Chern-Ricci flow for complex manifolds.
problem Understanding and solving open problems in Chern-Ricci flow.
method Parabolic flow of Hermitian metrics on complex manifolds.
result Open problems and new directions in Chern-Ricci flow highlighted.
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.
problem Proving a theorem about solutions to the quaternionic Monge-Ampère equation.
method Generalizing the parabolic Monge-Ampère equation to HKT geometry and proving existence and convergence of solutions.
result Existence and convergence of solutions to the equation under certain conditions.
Unified framework for Riemannian, Kähler, and quaternion-Kähler metrics.
problem Unified theories for Riemannian, Kähler, and quaternion-Kähler metrics.
method Projective parabolic geometries and classification of geometries.
result Unified framework generalizing classical theories.
Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant scalar curvature Kähler metrics on the iterated blowup are equivalent, for certain polarizations…
Study describes metrics on moduli space of Higgs bundles, proving exponential decay rate.
problem Analyzing the geometry of moduli space of Higgs bundles.
method Perturbing from approximate solutions to describe metrics, comparing to semi-flat metrics.
result Proves exponential decay rate of gL2−gsf=O(e−γt). Given a smooth complex projective variety X and a smooth divisor D on X, we prove the existence of Hermitian-Einstein connections, with respect to a Poincaré-type metric on X - D, on polystable parabolic principal Higgs bundles with parabolic structure over D, satisfying certain conditions on its restriction to D.
Local index theorem for cofinite hyperbolic Riemann surfaces derived from computational perspective.
problem Deriving the local index theorem for cofinite Riemann surfaces.
method Using Ahlfors' variational formulas and projection formulas, deriving integral formulas for variations of determinants.
result Explicit integral formulas for variations of logdetΔn and logdetNn. Heat kernels exist and are Hölder for rough metrics on smooth manifolds.
problem Existence and regularity of heat kernels on rough metrics.
method Local parabolic Harnack estimates for weak solutions in weighted Sobolev spaces.
result Globally continuous heat kernels are Hölder continuous locally.
In 1993, Bartnik introduced a quasi-spherical construction of metrics of prescribed scalar curvature on 3-manifolds. Under quasi-spherical ansatz, the problem is converted into the initial value problem for a semi-linear parabolic equation of the lapse function. The original ansatz of Bartnik started with a background …
A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to math.DG/0412405. We present a generalization of this construction to the case o…
The main result of this paper is a sufficient condition in order to have a compact Thom-Mather stratified pseudomanifold endowed with a c^-iterated edge metric on its regular part q-parabolic. Moreover, besides stratified pseudomanifolds, the q-parabolicity of other classes of singular spaces, such as compac…
The paper derives estimates and proves theorems for a specific type of nonlinear parabolic equation.
problem Analyzing solutions to a weighted nonlinear parabolic equation on metric measure spaces.
method Derives elliptic gradient estimates and proves Liouville-type theorems.
result Establishes conditions for the existence of positive ancient solutions.
We show that for two dimensional manifolds M with negative Euler characteristic there exists subsets of the space of smooth Riemannian metrics which are invariant and either parabolic or backwards-parabolic for the 2nd order RG flow. We also show that solutions exists globally on these sets. Finally, we establish the e…
The paper constructs stable Higgs bundles for hyperbolic metrics with singularities.
problem Existence of conformal hyperbolic metrics with prescribed singularities.
method Stable parabolic Higgs bundles of rank two.
result Alternative proof of Heins' theorem and extension of Hitchin's work.
Sharp Schauder estimates for conical Kähler metrics proved.
problem Developing Schauder estimates for equations with cone metrics.
method Proved sharp pointwise Schauder estimates for linear elliptic and parabolic equations on Cn with conical metrics. result Effective elliptic Schauder estimate and short time existence of conical Kähler-Ricci flow proved.
Paper introduces a new metric for deforming surfaces with parabolics.
problem Deformation spaces of quasifuchsian groups with parabolics.
method Developed a mapping class group invariant pressure metric on QF(S).
result Hausdorff dimension of limit sets varies analytically over QF(S).
New method constructs Ricci-flat metrics on Lie groups.
problem Finding Ricci-flat metrics on Lie groups.
method Combinatorial method to construct indefinite Ricci-flat metrics.
result Infinite families of Ricci-flat nilmanifolds constructed.
Study proves Hölder continuity for solutions to parabolic equations with conic singularities.
problem Proving Hölder continuity of solutions to parabolic equations with conic singularities.
method Using the results from previous works \cite{LSU} and \cite{FS}, the Hölder continuity is proven for linear parabolic equations of mixed type.
result Hölder continuity of solutions to parabolic equations with conic singularities is established.
In this paper, we introduce a new parabolic equation on Kähler manifolds. The static point of this flow is related to the existence of a lower bound of the Mabuchi energy. In this paper, we prove the flow always exists for all times for any initial smooth data. Further more, if the initial metric has non-negative bisec…
We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the underlying complex manifold. Solutions to the static equation are automatically H…
Strong bolicity helps prove Baum-Connes conjecture for certain hyperbolic groups.
problem Proving the Baum-Connes conjecture for relatively hyperbolic groups.
method Constructing a strongly bolic metric and using masks for random coset representatives.
result Deduced the Baum-Connes conjecture for groups satisfying (RD) and certain parabolics.
Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian-Yang-Mills equation for holomorphic vect…
We introduce a parabolic flow of almost Kahler structures, providing an approach to constructing canonical geometric structures on symplectic manifolds. We exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing our previous work on parabolic flows of Hermitian metrics. We e…
The paper consists of two parts. In the first one we show that a relatively hyperbolic group G splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups. As a corollary we obtain that every group which admits 3-discontinuous and 2-coc…
Geometric equation defines canonical metrics on vector bundle families.
problem Finding canonical metrics on families of holomorphic vector bundles.
method Introducing a geometric partial differential equation for families of holomorphic vector bundles.
result Construction of Hermite--Einstein metrics in adiabatic classes on product manifolds and proof of the existence of a unique solution for the Dirichlet problem.
The paper improves the description of Kähler metric flows and their singularities.
problem Improving the understanding of Kähler metric flows and their singularities.
method Parabolic regularizations of conjugate heat kernel potential functions based at almost-selfsimilar points.
result Tangent flows of Kähler metric flows admit nontrivial one-parameter actions by isometries.
We study a parabolic equation for finding solutions to the optimal transport problem on compact Riemannian manifolds with general cost functions. We show that if the cost satisfies the strong MTW condition and the stay-away singularity property, then the solution to the parabolic flow with any appropriate initial condi…
For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula co…