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 matrix Harnack inequalities for semilinear heat equations on manifolds.
problem Bounding solutions of semilinear heat equations on manifolds with geometric constraints.
method Applies Li-Yau estimates to derive Harnack inequalities for positive solutions.
result Derives matrix Harnack inequalities for positive solutions of semilinear heat equations.
Entropy-like energy decreases over time on special geometric manifolds.
problem Entropy-like energy dynamics on quaternionic contact and CR manifolds.
method Proof of monotonicity for heat equation.
result Entropy-like energy decreases over time on quaternionic contact and CR manifolds.
Geometric heat-flow theory detects Lagrangian coherent structures.
problem Detecting Lagrangian coherent structures in advective-diffusive systems.
method Transform Eulerian advection-diffusion to Lagrangian coordinates, then solve the resulting geometric heat equation.
result LCSs are boundaries of metastable sets under Lagrangian diffusion.
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. 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.
We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove the appropriate generalizations of Bochner--Weitzenböck formula and Laplacian comparison theorem, and study the h…
Graphs prove curvature condition with modified heat equation.
problem Proving curvature condition for infinite graphs.
method Establishing existence and uniqueness of modified heat equation solutions.
result Explicit examples of graphs satisfying assumptions.
The paper establishes estimates for heat equations and harmonic functions on metric spaces with curvature-dimension condition.
problem Analyzing geometric properties of metric measure spaces with curvature-dimension condition.
method Establishing local Li-Yau estimates and proving sharp Yau's gradient estimates for heat equations and harmonic functions.
result Sharp Li-Yau and gradient estimates for weak solutions of heat equations and harmonic functions on RCD∗(K,N) spaces. Study eigenvalues on quaternion-Kähler manifolds with geometric bounds.
problem Estimating eigenvalues on quaternion-Kähler manifolds.
method Lower bounds derived from modulus of continuity estimates for heat equation solutions and Laplace comparison theorem.
result Established bounds for first nonzero eigenvalues in terms of dimension, diameter, and scalar curvature.
Surveying progress on deformed Hermitian-Yang-Mills equation and its connections to stability.
problem Solvability of the deformed Hermitian-Yang-Mills equation and its relation to geometric stability.
method Utilizing geometric invariant theory (GIT) and Bridgeland stability theory to analyze the equation.
result On the blow-up of \(\mathbb{P}^2\), line bundles admitting a solution of the deformed Hermitian-Yang-Mills equation are Bridgeland stable, but not conversely.
The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold M with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to geometric flows. Secondly, we establish bounds for a solution ∇(t) of the Yan…
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.
Study describes heat kernel expansion for hypoelliptic operators.
problem Characterize coefficients in small time heat kernel expansion.
method Geometric characterization of coefficients using drift field and curvature-like invariants.
result Geometric characterization of coefficients in terms of drift field and curvature-like invariants.
Let M be a closed Riemannian manifold with a family of Riemannian metrics gij(t) evolving by a geometric flow ∂tgij=−2Sij, where Sij(t) is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-ty…
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…
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 "$…
Let M be a closed Riemannian manifold with a family of Riemannian metrics gij(t) evolving by geometric flow ∂tgij=−2Sij, where Sij(t) is a family of smooth symmetric two-tensors on M. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat …
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
Researchers compute heat kernel coefficients for 2D diffusion operators.
problem Analyzing heat kernel coefficients for 2D hypoelliptic operators.
method Explicit computation of heat kernel coefficients and interpretation in terms of curvature.
result Interpretation of heat kernel asymptotics for non-sub-Riemannian operators.
We give a proof of Gaussian upper bound for the heat kernel coupled with the Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type inequality that is conserved along Ricci ow. We will use a horizontal coupling of curve […
Geometric symbols help compute heat invariants.
problem Computing heat invariants efficiently.
method Geometric symbol calculus of pseudodifferential operators.
result Efficient computation of heat invariants.
In this paper we introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution of this flow, discuss its convergence and possible applications, and its relation to the …
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.
Quaternionic contact heat equation studied on compact manifolds.
problem Heat equation on quaternionic contact manifolds.
method Introduced quaternionic contact heat equation and energy functional.
result Monotonicity of qc energy functional along the heat equation.
New heat equation method solves intertwining problems in CR geometry.
problem Intertwining problems in conformal CR geometry.
method Heat equation and extension problems approach.
result New intertwining formulas derived.
Develops a local Fokker--Planck geometric framework for more accurate score estimation.
problem Inaccurate estimation of score function in non-linear, state-dependent drifts.
method Local Fokker--Planck geometric framework, time change to cumulative-variance coordinate, heat-ball mean-value representations, exact high-dimensional sampling.
result Exact local mean-value representations for the score and density, improved accuracy in low-density regions.
Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.
problem Estimating solutions to CR heat equation on complex manifolds.
method Local and global Li-Yau type gradient estimates.
result Gradient estimates and Harnack inequality for positive solutions.
Proposes learning manifold implicitly via heat kernel.
problem Direct manifold learning methods lack flexibility for down-stream applications.
method Implicit manifold learning using heat kernel.
result Framework achieves state-of-the-art results for data generation and Bayesian inference.
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.
Sharp time analyticity proved for heat equation on Ricci solitons.
problem Analyticity of heat equation solutions on gradient shrinking Ricci solitons.
method Proved analyticity for solutions with quadratic exponential growth.
result Sharp growth condition for analyticity is established.
This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.
problem Investigating symmetries of biharmonic heat equations on surfaces of revolution.
method Lie symmetry analysis to classify symmetries and derive invariant solutions.
result The biharmonic heat equation on a surface of revolution has the same Lie symmetries as the harmonic heat equation.
Study nonlinear heat equation gradient estimates and applications.
problem Gradient estimates for nonlinear heat equation.
method Elliptic gradient estimates for a nonlinear f-heat equation. result Obtain gradient estimates for positive solutions.
In this paper, we study the Poisson equation and heat equation in a model matrix geometry Mn. Our main results are about the Poisson equation and global behavior of the heat equation on Mn. We can show that if c0 is the initial positive definite matrix in Mn, then c(t) exists for all time and is positive …
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…
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.
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.
This paper proposes an unsupervised learning method to solve heat equations on chips.
problem Critical need for solving heat transfer equations on chips for 5G and AI.
method Hybrid framework of Auto Encoder and Image Gradient for unsupervised learning.
result Framework can solve heat transfer problems with a single training process and predict unseen cases.
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.
Ancient solutions to heat equations on graphs are shown to be analytic in time.
problem Analyzing the analyticity of ancient solutions to heat equations on graphs.
method Proving time analyticity under a sharp growth condition.
result Ancient solutions to heat equations on graphs are time analytic under certain conditions.
We construct examples of 2-step Carnot groups related to quaternions and study their fine structure and geometric properties. This involves the Hamiltonian formalism, which is used to obtain explicit equations for geodesics and the computation of the number of geodesics joining two different points on these groups. We …
Study on biharmonic heat equation on manifolds with curvature constraints.
problem Analyzing entire solutions of biharmonic heat equation on manifolds.
method Exponential decay estimates for biharmonic heat kernel under Ricci curvature and noncollapsing conditions. Proving uniqueness criteria for Cauchy problem.
result Conservation law for biharmonic heat kernel and uniform L-infinity estimate for entire solutions.
Proves uniqueness of heat equation solutions on Riemannian manifolds.
problem Proving uniqueness of solutions to the heat equation on Riemannian manifolds.
method Analyzes Lp solutions for 0<p<1 and improves L1 uniqueness result. result Improves curvature assumption for L1 uniqueness result. Heat semigroups used to solve geometric inequalities on manifolds.
problem Finding geometric inequalities on Riemannian and sub-Riemannian manifolds.
method Heat semigroups techniques applied to Riemannian and sub-Riemannian geometry.
result Applications of heat semigroups in geometric inequalities.
Ancient solutions of heat equation with exponential growth are analytic in time.
problem Analyticity of solutions to the heat equation in time.
method Proving analyticity for ancient solutions with exponential growth.
result Ancient solutions with exponential growth are analytic in time.
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 …
Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.
problem Bounding ancient solutions to biharmonic heat equation.
method Using polynomial volume growth and dimensions of biharmonic functions.
result Ancient solutions are bounded by polynomial dimensions.
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…