Bounds on Hessian of heat equation coupled with Ricci flow.
problem Estimating the Hessian of a solution to the conjugate heat equation coupled with Ricci flow.
method Obtained upper bounds for the Hessian.
result Local and global upper bounds for the Hessian of a positive solution.
Article proves Liouville theorem for heat equation in super Ricci flow.
problem Proving Liouville theorem for heat equation in super Ricci flow.
method Formulated under a growth condition concerning Perelman's reduced distance.
result Established Liouville theorem for heat equation in ancient super Ricci flow.
We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the ε-Ricci flow on a closed surface. These new Harnac…
Extends gradient estimates for heat equation under Finsler geometric flows.
problem Global gradient estimates for positive solutions to heat equation.
method General compact Finsler CD(−K,N) geometric flow. result Derives Harnack inequality for positive solutions.
Derives heat equation estimates linked to Ricci flow on compact and noncompact manifolds.
problem Estimating heat equation coupled to Ricci flow on noncompact manifolds.
method Local derivative estimates for the heat equation coupled to the Ricci flow.
result Extends results on distance distortion and backward pseudolocality to noncompact manifolds.
In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's …
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.
Paper studies heat flow for maps on manifolds, avoiding singularities.
problem Avoiding singularities in heat flow for maps on manifolds.
method Introduces regularized conformal heat flow for n-harmonic maps. result Regularized n-conformal heat flow does not develop finite time singularities. 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.
Proves estimates for Kähler-Ricci flow solutions.
problem Positive solutions to Kähler-Ricci flow.
method Matrix Li-Yau-Hamilton estimates coupled with flow.
result Monotonicity formula derived.
Study heat flows on time-dependent metric measure spaces, proving properties related to super-Ricci flows.
problem Characterize heat flows and their properties on time-dependent metric measure spaces.
method Prove existence, uniqueness, and regularity of heat equations and their duals on time-dependent metric measure spaces.
result Equivalence of dynamic convexity of Boltzmann entropy, monotonicity of Wasserstein distances, gradient estimates, and Bochner inequality.
Estimates heat equation on shrinking Ricci solitons with uniform bounds.
problem Analyzing heat equation on shrinking Ricci solitons.
method Proved L2 estimate with time-dependent Gaussian weight. result Uniform bounds for heat equation along Ricci flow.
Survey on heat equation estimates on manifolds.
problem Estimating heat equations on manifolds.
method Recalling and discussing Li-Yau, Hamilton, Perelman's estimates and their applications.
result Sharp constants and improved curvature conditions for heat equations on manifolds.
New entropy formulae for heat equation on manifolds.
problem Entropy formulae for linear heat equation on Riemannian manifolds.
method Proved new entropy formulae for linear heat equation on static Riemannian manifolds with nonnegative Ricci curvature.
result Results are analogies of Cao and Hamilton's entropies for Ricci flow.
Heat and entropy flows linked in Carnot groups.
problem Understanding heat and entropy dynamics in Carnot groups.
method Proving correspondence between heat equation solutions and gradient flows of entropy in Wasserstein space.
result Complete answer to a question left open by N. Juillet.
The paper proves new Harnack inequalities for various nonlinear heat equations on manifolds.
problem Analyzing and proving new Harnack inequalities for nonlinear heat equations.
method Proving constrained trace, matrix, and interpolated Harnack inequalities for specific nonlinear heat equations.
result Derives new differential Harnack inequalities with time-exponential correction terms.
Paper approximates backward heat equation using wave equations and Ricci flow.
problem Solving backward heat equation on manifolds using wave equations.
method Approximates solutions of a wave equation on a larger manifold with Ricci flow to solve the backward heat equation.
result The approximation provides solutions to the backward heat equation on manifolds.
The Levy Laplacian is studied on manifolds, with heat flow solutions tending to constant functionals over time.
problem Understanding the Levy Laplacian on manifolds and its heat flow behavior.
method Various definitions of the Levy Laplacian are proven equivalent. Heat flows of differential forms are used to construct solutions.
result Solutions of the heat equation with the Levy Laplacian tend to locally constant functionals over time.
The paper extends Li-Yau-Hamilton estimates to evolving Kähler metrics and nonlinear heat equations.
problem Deriving estimates for nonlinear heat equations on evolving Kähler metrics.
method Generalized matrix Li-Yau-Hamilton estimates to Kähler manifolds with evolving metrics and nonlinear heat equations.
result Extended Li-Yau-Hamilton estimates to evolving Kähler metrics and nonlinear heat equations.
The paper derives Harnack inequalities for positive heat equation solutions on Finsler manifolds.
problem Deriving inequalities for heat equation solutions on Finsler manifolds.
method Generalizing Li-Yau type gradient estimates to Finsler geometry and applying to heat equation solutions.
result General gradient estimate for positive solutions of the heat equation on Finsler manifolds under curvature assumptions.
In this paper, we derive a Sobolev inequality along an extended Ricci flow and prove a point-wise Guassian type bound for the fundamental solutions of the conjugate heat equation under the flow.
The paper studies Gauss maps of a Ricci-mean curvature flow.
problem Investigating the Gauss maps of a Ricci-mean curvature flow.
method Deduced the evolution equation for the Gauss maps of a Ricci-mean curvature flow and proved they satisfy the vertically harmonic map heat flow equation.
result The Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation.
The paper proves stability of a blowup solution for Yang-Mills heat flow.
problem Stability of blowup solutions for Yang-Mills heat flow.
method Small perturbation analysis and explicit self-similar blowup solution.
result Stability of the explicit self-similar blowup solution under perturbations.
The paper proves entropy power properties on Riemannian manifolds and Ricci flows.
problem Entropy power on Riemannian manifolds and Ricci flows.
method Proving concavity and convexity of Shannon entropy power for heat and conjugate heat equations on Riemannian manifolds and Ricci flows.
result Entropy power rigidity models on Einstein or quasi Einstein manifolds and shrinking Ricci solitons.
Two graphs show mean curvature flow differs from heat flow in dimensions n≥2.
problem Comparing mean curvature flow and heat flow on entire graphs.
method Analyzing two specific graphs in dimensions n≥2.
result Mean curvature flow and heat flow behave differently, with oscillation vs. stabilization.
Derives gradient estimation for a specific heat equation on evolving manifolds.
problem Gradient estimation for a generalized heat equation on evolving weighted Riemannian manifolds.
method Derives gradient estimation for a specific heat equation on evolving weighted Riemannian manifolds.
result Derives a Harnack type inequality and a Liouville type theorem as applications of gradient estimation.
Second part of a study on heat equations on special manifolds, focusing on parametrix construction.
problem Analysis of heat-type equations on manifolds with fibered boundaries.
method Construction of parametrix for heat-type equations.
result Inference of existence and regularity of certain parabolic equations.
The paper considers a manifold M evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on M. Among other results, we prove Li-Yau-type inequalities in this context. We consider both the case where M is a complete manifold without boundary and the …
Estimates for Dirac solutions applied to a new elliptic-parabolic problem.
problem Solving Dirac equations with specific boundary conditions.
method Developed estimates and derived existence and uniqueness results.
result General existence, uniqueness, and regularity theorem for Dirac equations.
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.
In this paper, we study the partial convexity of smooth solutions to the heat equation on a compact or complete non-compact Riemannian manifold M or Kahler-Ricci flow. We show that under a natural assumption, a new partial convexity property for smooth solutions to the heat equation is preserved.
The paper proves Harnack inequalities on foliations with Ricci flow.
problem Proving Harnack inequalities on foliations with Ricci flow.
method Using transverse Ricci flow on totally geodesic Riemannian foliations, the paper proves two types of differential Harnack inequalities and a time-dependent curvature dimension inequality.
result The paper establishes Harnack inequalities and heat kernel upper bounds.
We present two approaches to the heat flow on a Finsler manifold (M,F): either as gradient flow on L2(M,m) for the energy; or as gradient flow on the reverse L2-Wasserstein space P2(M) of probability measures on M for the relative entropy. Both approaches depend on the choice of a measure m on …
In this paper, we study how to get the Ricci expanders from W+-functional through the heat kernel estimate of the conjugate heat equation to the type III singularity of Ricci flow. The Gaussian upper and lower bounds are established for the related heat kernel in accordance to the interesting work of Cao-Zhang for the …
We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's pr…
Study on heat flow across two half-lines with special boundary conditions.
problem Low energy mode of heat flow transmission across a Grushin-type cylinder.
method Analysis of heat equation with inverse-square potential and bridging boundary conditions.
result First insight into qualitative features of the heat flow solution at later times.
Study fast diffusion equation under Ricci flow with estimates.
problem Analyzing fast diffusion equation under Ricci flow.
method Proved Aronson-Bénilan and Li-Yau-Hamilton type estimates.
result Extended Li-Yau-Hamilton estimates to noncompact settings.
A combinatorial version of Yamabe flow is presented based on Euclidean triangulations coming from sphere packings. The evolution of curvature is then derived and shown to satisfy a heat equation. The Laplacian in the heat equation is shown to be a geometric analogue of the Laplacian of Riemannian geometry, although the…
In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem ut−Δu=aulogu, u>0 on the compact Riemannian manifold (M,g) of dimension n and with non-negative (Bakry-Emery)-Ricci curvature. Here…
New approach to heat flow for half-harmonic maps, related to minimal surfaces.
problem Heat flow for half-harmonic maps from S1 to closed target manifolds. method Classical approach using Dirichlet-to-Neumann operator for the Laplace equation.
result Analogous results to 1985 harmonic map heat flow, valid for finite-energy data.
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.
The paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold M evolving under the Ricci flow, coupled with the harmonic map flow between M and a second manifold N. We prove Li-Yau type Harnack inequalities and we consider the cases when M is a complete manifold …
Paper proves Harnack inequalities for Witten Laplacian on manifolds with specific flows.
problem Proving Harnack inequalities for Witten Laplacian on Riemannian manifolds.
method Using Li-Yau and Hamilton type inequalities for heat equation associated with time-dependent Witten Laplacian on manifolds with specific flows.
result Proves Li-Yau and Hamilton type Harnack inequalities for Witten Laplacian.
Novel heat flow estimates on ALE manifolds for Schrödinger operators.
problem Estimating heat flows on ALE manifolds with non-trivial L2-kernel. method Combining Fredholm theory for Dirac type operators and heat kernel advances.
result Established Lp−Lq decay estimates for heat flows. In this paper we prove first order differential Harnack estimates for positive solutions of the heat equation (in the sense of distributions) under closed Finsler-Ricci flows. We assume mild non-linearities (in terms of the Chern connection, S−curvature and Hessian) and suitable Ricci curvature bounds throug…
We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…
Study proves short-term existence for harmonic maps under evolving metrics.
problem Analyzing harmonic maps under time-dependent metrics.
method Proves short-term existence for harmonic map heat flow coupled with a smooth family of complete metrics.
result Generalizes short-term existence results for harmonic map heat flow.
In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in Bk×(0,T) to the minimal fundamental solut…