arXiv research

New rigidity results for a generalized Ricci-Hessian equation on manifolds.

Estimates heat equation on shrinking Ricci solitons with uniform bounds.

We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…

Bounds on Hessian of heat equation coupled with Ricci flow.

Solves Calabi-Yau equation on complex manifolds with non-positive astheno-Ricci curvature.

The paper improves heat equation estimates under weaker Ricci curvature conditions.

The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.

The paper studies a degenerate equation related to Kähler-Ricci flow on symplectic quotients.

Article proves Liouville theorem for heat equation in super Ricci flow.

The paper examines conditions for Ricci solitons to be Ricci flat or Einstein.

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 $\varepsilon$-Ricci flow on a closed surface. These new Harnac…

The fundamental equations of Gauss, Codazzi and Ricci provide the conditions for local isometric embeddability. In general, the three fundamental equations are independent for surfaces in Riemannian 4-manifolds. In contrast, we prove in this article that for arbitrary Lorentz surfaces in Lorentzian Kaehler surfaces the…

Proves estimates for Kähler-Ricci flow solutions.

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve t…

The purpose of this paper is to introduce the Ricci Yang-Mills soliton equations on nilpotent Lie groups. In the 2-step nilpotent setting, we show that these equations are strictly weaker than the Ricci soliton equations. Using techniques from Geometric Invariant Theory, we develop a procedure to build many different k…

New classification of 5D nilsolitons using algebraic Ricci soliton equation.

Gradient estimates for a parabolic PDE under Ricci-Bourguignon flow on warped product manifolds.

In this paper, the author has considered the hyperbolic Kahler-Ricci flow introduced by Kong and Liu [11], that is, the hyperbolic version of the famous Kahler-Ricci flow. The author has explained the derivation of the equation and calculated the evolutions of various quantities associated to the equation including the…

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…

Flow approach solves Ricci equation boundary problem.

We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the role that conformal geometry plays in constraining the scalar curvature. These equ…

Study expanding Ricci solitons on vector bundles, reducing to Higgs bundle equations.

$\mathscr{W}$-entropy and reduced volume for the Ricci flow were introduced by Perelman, which had proved their importance in the study of the Ricci flow. L. Ni studied the analogous concepts for the linear heat equation on the static manifolds, and established an equation which links the large time behavior of these t…

Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and…

The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.

The paper characterizes Ricci solitons on the Poincaré upper half plane.

Derives gradient bounds for f-heat equations on manifolds with Bakry-Emery Ricci curvature.

Paper defines parabolic frequency for Ricci flow solutions, proving monotonicity and uniqueness.

Paper extends Aronson-Bénilan estimates for porous medium equations on manifolds with negative curvature.

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 …

Solves complex Monge-Ampère equation for Kähler-Ricci solitons.

Estimates for solutions on manifolds under Ricci flow.

We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. Some simple proofs of known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended thi…

Paper proves estimates for heat and conjugate heat equations under Ricci flow, leading to monotonicity of parabolic frequencies.

New families of Ricci solitons found with collapsing volume.

Enhances understanding of Kähler-Ricci flow singularities.

The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.

The paper proves entropy power properties on Riemannian manifolds and Ricci flows.

The article proves Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.

This paper studies gradient almost Ricci-harmonic soliton with respect to a fixed metric. We rely on analytic techniques to estabilish some basic elliptic and integral equations for the structure of almost Ricci-harmonic soliton which generalizes that of Ricci-hamonic solitons on one hand and that of almost Ricci solit…

The study explores special Ricci-Hessian equations on Kähler manifolds and identifies three types of solutions.

We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equa…

The study finds infinite nodal solutions for equations on positive Ricci curvature manifolds.

In this article we study the Kähler Ricci flow, the corresponding parabolic Monge Ampère equation and complete non-compact Kähler Ricci flat manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$ is sufficiently close to being Kähler Ricci flat in a suitable sense, then the Kähler Ricci flow \eqr…

Gradient bounds and Liouville theorems for quasi-linear equations on manifolds with nonnegative Ricci curvature.

Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduc…

In this paper, we study the class of Finsler metrics, namely (α, β)- metrics, which satisfies the un-normal or normal Ricci flow equation.