Sharp convergence theorem for Yang-Mills flow on ALE manifolds proved.
problem Proving convergence of Yang-Mills flow on ALE gravitational instantons.
method Noncompact version of the 'parabolic gap theorem'.
result Sharp convergence theorem for Yang-Mills flow on ALE 4-manifolds.
The paper proves parabolic gap theorems for Yang-Mills energy.
problem Yang-Mills energy and instantons on various manifolds.
method Parabolic Yang-Mills flow and Morrey norms.
result Spaces of connections with Yang-Mills energy less than a certain threshold deformation-retract onto spaces of instantons.
The paper proves gap results for self-shrinkers in r-mean curvature flow.
problem Understanding the gap in properties of self-shrinkers in r-mean curvature flow. method Proving gap results using a modified second fundamental form and a differential operator.
result Proper self-shrinkers are parabolic for a certain second-order differential operator.
The paper proves smoothness of transition layers in the Allen-Cahn equation.
problem Proving uniform C2,α regularity for transition layers. method Utilizes Allen-Cahn monotonicity formula, Lipschitz approximation, and blowups.
result Shows uniform C2,α regularity for transition layers converging to smooth mean curvature flows. 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.
problem Proving nonexistence results for parabolic inequalities on Riemannian manifolds.
method Using a test function argument and weighted volume growth assumptions.
result Established Liouville-type theorems for (p,q)-Laplacian operator inequalities. 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.
problem Determining the next realizable symmetry dimension in parabolic geometries.
method Analyzing submaximally symmetric structures of type (G,P) for specific Lie groups. result Local uniqueness of submaximally symmetric structures established.
Proves monotonicity of parabolic frequency on all manifolds without curvature assumptions.
problem Monotonicity of parabolic frequency on manifolds.
method Analyzes parabolic frequency function on manifolds, proving monotonicity without curvature assumptions.
result Monotonicity of parabolic frequency on all manifolds, no curvature assumption needed.
Generalized Agol's theorem to 3-manifold groups.
problem Two-parabolic-generator subgroups in hyperbolic 3-manifold groups.
method Generalization of Agol's proof for 2-bridge link groups to 3-manifold groups.
result Refinement of Boileau-Weidmann's result.
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.
problem Exploring new phenomena in boundaries of relatively hyperbolic groups.
method Combination theorem to create examples of relatively hyperbolic groups with parabolic cut pairs.
result All relatively hyperbolic groups with inseparable parabolic cut pairs arise via the combination theorem.
Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.
problem Estimating gradients for nonlinear parabolic equations on Riemannian manifolds.
method Analyzes Fisher-KPP, parabolic Allen-Cahn, and Newell-Whitehead equations on complete noncompact Riemannian manifolds.
result Gradient estimates for positive solutions and Liouville theorem for ancient solutions.
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.
problem Defining and analyzing non-parabolic spatial hybrid framed curves.
method Definition and proof of existence and uniqueness theorem for non-parabolic spatial hybrid framed curves.
result Existence and uniqueness theorem for non-parabolic spatial hybrid framed curves.
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.
problem Analyzing the behavior of random walks on relatively hyperbolic groups.
method Study of convergent random walks with finite derivative of Green function at spectral radius.
result Proves a local limit theorem for the probability of returning to the origin.
Maximal regularity for nonuniformly parabolic problems with normal degeneration.
problem Nonuniformly parabolic boundary value problems with degeneration in normal direction.
method Theory of linear parabolic differential equations on noncompact Riemannian manifolds.
result Optimal solution theory for natural degeneration case.
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.
The paper defines capacities for minimal graphs over manifolds and proves the half-space property.
problem Characterizing minimal graphs and their properties over manifolds.
method Defining capacities using relative volume, studying solutions of bounded variation, and analyzing boundary behavior.
result Proves the half-space property for M-parabolic manifolds. The trace set of a Fuchsian group Γ ist the set of length of closed geodesics in the surface Γ\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.
problem Frequency monotonicity for positive solutions of nonlinear equations under Ricci flow.
method Obtained parabolic frequency monotonicity for solutions of two nonlinear parabolic equations with bounded Ricci curvature.
result Established integral type Harnack inequalities using parabolic frequency monotonicity.
The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.
problem Mapping and identifying symplectic structures of two types of hyperkähler manifolds.
method Produced a map from star-shaped quiver varieties to Higgs bundle moduli spaces, verified stability, and showed it is a homeomorphism.
result Identified natural holomorphic symplectic structures on the two spaces.
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.
problem Proves properties of complex braid group parabolic subgroups.
method Uses Garside groupoid structure of B(G31) to extend results.
result Proves main theorems for B(G31) parabolic subgroups.
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. The paper extends Huber's theorem to higher dimensions using n-Laplace equations.
problem Proving finite point conformal compactification for general dimensions.
method Using n-Laplace equations and strengthened Arsove-Huber's theorem.
result Established finite point conformal compactification theorem for manifolds.
Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.
Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.
problem Derives gradient estimate for a nonlinear parabolic equation on Finsler manifolds.
method Leverages a new Laplacian comparison theorem to derive a Li-Yau type gradient estimate.
result Establishes a Li-Yau type gradient estimate for the Finslerian logarithmic Schrödinger equation.
Let M2 be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in M2×R. First, using a characterization of δ-parabolicity we prove that under additional conditions on 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.
problem Mean curvature flow and its parabolic analogue.
method Improved convergence property and curvature estimate.
result Curvature estimate for parabolic Allen-Cahn equation.
Researchers derived heat kernel expansions for non-compact spaces using Witten deformation.
problem Heat kernel expansions on non-compact spaces, especially for Witten Laplacians.
method Introduced parabolic distance and used it to derive asymptotic expansions.
result Derived an asymptotic expansion of trace of heat kernel for small-time t. The paper extends a theorem about stable minimal surfaces to higher codimensions.
problem Stability and holomorphicity of parabolic stable minimal surfaces in higher-dimensional spaces.
method Generalization of a classical theorem to higher codimensions, with additional assumptions on the normal bundle.
result Holomorphicity of stable minimal surfaces in higher-dimensional spaces.
The paper proves gap theorems for Yang-Mills on manifolds with positive Yamabe.
problem Yang-Mills theory on manifolds with positive Yamabe constant.
method Extending Gursky-Kelleher-Streets results to complete manifolds.
result Equality in gap theorem described in terms of basic instanton.
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.
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.
problem Estimating solutions to nonlinear weighted parabolic equations.
method Derives Li-Yau and Hamilton type gradient estimates, and Hessian estimates.
result New gradient and Hessian estimates for positive solutions of 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.
problem Index theory of curved Bernstein-Gelfand-Gelfand sequences.
method Utilizes K-homology and noncommutative geometry.
result Solves the index problem for BGG-sequences on flat parabolic geometry.
The paper solves a complex financial optimization problem using a novel mathematical technique.
problem Optimizing portfolio selection in financial markets.
method Maximal monotone operator method and Riccati transformation.
result Existence and uniqueness of a solution to the transformed parabolic equation in a Sobolev space.
Combination theorem for PGF groups helps in constructing new examples and understanding their geometry.
problem Understanding the geometry of PGF groups and their combinations.
method Utilizing subsurface projection to control the geometry of fundamental groups of graphs of PGF groups.
result Combination theorem for PGF groups and other generalizations.