The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.
problem Understanding the behavior of parabolic frequency under Ricci flow and Ricci-harmonic flow.
method Investigates the monotonicity of parabolic frequency for solutions of linear and heat equations with bounded curvatures.
result Establishes monotonicity results for parabolic frequency under specific curvature conditions.
Paper defines parabolic frequency for Ricci flow solutions, proving monotonicity and uniqueness.
problem Backwards uniqueness for solutions of parabolic equations on Ricci flows.
method Defines and proves monotonicity of parabolic frequency for Ricci flow solutions.
result Backwards uniqueness for solutions of parabolic equations on Ricci flows.
The paper defines a frequency for mean curvature flow and proves its monotonicity.
problem Backwards uniqueness for solutions of mean curvature flow.
method Defining and proving monotonicity of a parabolic frequency for mean curvature flows.
result Frequency monotonicity implies backwards uniqueness for mean curvature flow solutions.
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.
Ansatze are constructed under which the solutions of 11D supergravity must be stationary points of a parabolic flow on a Riemannian manifold M10−p. This parabolic flow turns out to be the Ricci flow coupled to a scalar field, a (3−p)-form, and a 4-form. This allows the introduction of techniques from parabolic…
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.
New criteria for long-time existence of parabolic flow from 11D supergravity.
problem Establishing long-time existence of a parabolic flow from 11D supergravity.
method Using only Ricci curvatures and their derivatives, along with 4-forms, to establish long-time existence.
result A new criteria for long-time existence of the parabolic flow.
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.
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…
Paper proves estimates for heat and conjugate heat equations under Ricci flow, leading to monotonicity of parabolic frequencies.
problem Establishing estimates for heat and conjugate heat equations under Ricci flow.
method Proving matrix Li-Yau-Hamilton estimates for positive solutions to the heat and conjugate heat equations coupled with Ricci flow.
result Monotonicity of parabolic frequencies established up to correction factors.
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.
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…
Gradient estimates for a parabolic PDE under Ricci-Bourguignon flow on warped product manifolds.
problem Analyzing the Ricci-Bourguignon flow on warped product manifolds.
method Establishing gradient estimates for a parabolic partial differential equation.
result Gradient estimates for the parabolic PDE provide analytic input for geometric applications.
The paper estimates gradients for a weighted parabolic equation under geometric flow.
problem Estimating gradients for a specific parabolic equation on a weighted manifold.
method Obtained space-time gradient estimates through integrating the equation.
result Found corresponding Harnack inequalities through gradient estimates.
Brakke flow support is parabolically rectifiable
problem Support of Brakke flow is parabolically rectifiable
method Developed approach to Brakke flow as space-time-Grassmann measure
result Standard convergence of Brakke flows is equivalent to space-time-Grassmann Radon measures
We generalize the classical Bochner formula for the heat flow on evolving manifolds (M,gt)t∈[0,T] to an infinite-dimensional Bochner formula for martingales on parabolic path space PM of space-time M=M×[0,T]. Our new Bochner formula and the inequalities that follow from it a…
We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…
New non-canonical flows found via parabolic Allen-Cahn equations.
problem Existence of non-canonical mean curvature flows inside fattening regions.
method Construction of non-canonical flows as limits of parabolic ε-Allen-Cahn solutions.
result First examples of non-outermost, non-canonical integral Brakke motions.
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…
The paper studies gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
problem Gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
method Gradient estimates and Harnack inequalities for heat equations under the Laplacian G_2 flow.
result Monotonicity of parabolic frequency and backward uniqueness for positive solutions.
In this paper we find solutions uε to a certain class of vector-valued parabolic Allen-Cahn equation that as ε→0 develops as interface a given triod evolving under curve shortening flow.
Study geodesic flow on symmetric surfaces to determine parabolic type.
problem Determine conditions for a Riemann surface to be of parabolic type.
method Analyze Fenchel-Nielsen coordinates and covering group properties.
result Conditions for a surface to be parabolic are equivalent to specific properties of its Fenchel-Nielsen coordinates.
In this paper, by maximum principle and cutoff function, we investigate gradient estimates for positive solutions to two nonlinear parabolic equations under Ricci flow. The related Harnack inequalities are deduced. An result about positive solutions on closed manifolds under Ricci flow is abtained. As applications, gra…
Paper solves Musielak-Orlicz-Gauss image problem using parabolic flows.
problem Characterize Musielak-Orlicz-Gauss image measure of convex bodies.
method Study of parabolic flows and approximation technique.
result Provides solutions to the extended Musielak-Orlicz-Gauss image problem.
We study the parabolic flow for generalized complex Monge-Ampère type equations on closed Hermitian manifolds. We derive {\em a priori} C∞ estimates for normalized solutions, and then prove the C∞ convergence.
Characterizes parabolic flute surfaces with specific parameters.
problem Characterizing flute surfaces with ergodic geodesic flow.
method Extending results on Fenchel-Nielsen coordinates and analyzing twist parameters.
result Characterizes parabolic flute surfaces with twist parameters in {0,1/2}.
Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.
problem Understanding the Kobayashi-Hitchin correspondence for specific sheaves.
method Using Hermitian-Yang-Mills flow on Kähler manifolds with simple normal crossing divisors.
result Established the correspondence for saturated reflexive parabolic sheaves.
The paper improves heat equation estimates under weaker Ricci curvature conditions.
problem Improving heat equation estimates under weaker Ricci curvature conditions.
method Establishing Li-Yau-type and Hamilton-type estimates for positive solutions of the heat equation under generalized Ricci flow.
result Deriving Harnack-type inequalities and monotonicity of parabolic frequency.
Study shows long-term flow on special manifolds with positive Yamabe constant.
problem Analyzing long-time behavior of Yamabe flow on singular spaces.
method Formulated axioms for long-time existence, used parabolic Moser iteration for bounds.
result Established long-time existence of normalized Yamabe flow on specified manifolds.
The paper provides gradient estimates for a parabolic equation under Finsler geometric flows.
problem Gradient estimates for a general parabolic equation under compact Finsler CD(−K,N) geometric flows. method Presented Shi-type and Hamilton-type gradient estimates.
result Demonstrates the possibility of removing stricter derivative bounds imposed by Finsler curvature conditions.
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.
A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.
Non-ergodic geodesic flow on Cantor tree surfaces found.
problem Determining when geodesic flow on Cantor tree surfaces is non-ergodic.
method Interpolating between two rates of convergence of cuff lengths to zero to prove non-ergodicity.
result Cantor tree surfaces with certain rates of cuff length convergence are non-parabolic.
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…
This article studies a discrete geometric structure on triangulated manifolds and an associated curvature flow (combinatorial Yamabe flow). The associated evolution of curvature appears to be like a heat equation on graphs, but it can be shown to not satisfy the maximum principle. The notion of a parabolic-like operato…
Researchers prove twists can make surfaces parabolic even with large cuff lengths.
problem Making surfaces parabolic with large cuff lengths and twists.
method Choosing lengths and applying twists to make surfaces parabolic.
result For any sequence of positive numbers, there is a choice of lengths such that twists make the surface parabolic.
We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we sho…
The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
problem Mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
method Analyzes the parabolic equation and Monge-Ampère type equation, proving smooth solutions and convergence to self-expanding solutions.
result Smooth solutions u(x,t) for specific nonlinear equations and convergence to self-expanding solutions. Paper studies flows of spinor fields with flux for unified theories.
problem Existence of covariantly constant spinors in unified theories.
method Introduces parabolic flows of spinor fields to find stationary points.
result Establishes short-time existence and smoothing estimates for spinor flows.
We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.
In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equation \[ (Δ-q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this …
The first part of the paper discusses a second-order quasilinear parabolic equation in a vector bundle over a compact manifold M with boundary ∂M. We establish a short-time existence theorem for this equation. The second part of the paper is devoted to the investigation of the Ricci flow on M. We propose …
Proves Thom's conjecture for parabolic flows on Hilbert spaces.
problem Gradient flows on infinite-dimensional spaces and geometric flows with symmetry.
method Analytic functions, Hilbert spaces, Yang-Mills Flow, Ricci flow, critical points, Lojasiewicz inequality.
result Gradient conjecture holds for parabolic flows on Hilbert spaces, including flows with gauge symmetry.
Study curves evolving on hypersurfaces with free boundaries, preserving length.
problem Evolution of curves on hypersurfaces with free boundaries.
method Nonlocal evolution equation with nonlinear boundary conditions, short-time existence, uniqueness, and parabolic energy estimates.
result Global existence and convergence to critical points proved.
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…
Solves long-time solutions for a specific equation on hyperkähler manifolds.
problem Finding solutions to a specific equation on hyperkähler manifolds.
method Introduced a parabolic quaternionic Monge-Ampère equation and proved its long-time solvability.
result Smooth convergence to a solution of the quaternionic Monge-Ampère equation.
Study curve shortening flow on Riemann surfaces with conical singularities.
problem Evolution of curves on Riemann surfaces with singular points.
method Curve shortening flow governed by a degenerate quasilinear parabolic equation.
result Evolving curves stay fixed at singular points and show collapsing and convergence results.
In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-…