Related papers: A Bochner Formula on Path Space for the Ricci Flow

In this paper we survey the recent developments of the Ricci flows on complete noncompact K\"{a}hler manifolds and their applications in geometry.

We prove that, for a two-dimensional Riemannian manifold, the Ricci flow is obtained by a Wiener process.

We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M,g) such that: a) (M,g) is non-collapsed, b) the Ricci curvature of (M,g) is bounded from below, c) the geometry of (M,g) at infinity is not too extreme.…

We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are…

We study Ricci flows of some classes of physically valuable solutions in Einstein and string gravity. The anholonomic frame method is applied for generic off-diagonal metric ansatz when the field/ evolution equations are transformed into…

We study relation of the Ricci Flow on 3-dimensional Lie groups and 4-dimensional Ricci-flat manifolds. In particular, we construct Ricci-flat cohomogeneity one metrics with respect to 3-dimensional Lie groups.

A geometric flow based in the Riemann-Christoffel curvature tensor that in two dimensions has some common features with the usual Ricci flow is presented. For $n$ dimensional spaces this new flow takes into account all the components of the…

In this note, we want to establish several formulas about functionals along harmonic Ricci flow on surface with boundary

We study the long time behaviour of Ricci flow with bubbling-off on a possibly noncompact $3$-manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston's hyperbolisation…

This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and K\"ahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and…

This article considers nonlocal heat flows into a singular target space. The problem is the parabolic analogue of a stationary problem that arises as the limit of a singularly perturbed elliptic system. It also provides a gradient flow…

The examples of the Ricci flows on four-dimendionsl manifolds which are determined by help of nonlinear differentials equations of the type of Monge-Ampere are constructed. Their particular solutions and their properties are discussed.

Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various…

In this paper, we construct a pyramid Ricci flow starting with a complete Riemannian manifold $(M^n,g_0)$ that is PIC1, or more generally satisfies a lower curvature bound $K_{IC_1}\geq -\alpha_0$. That is, instead of constructing a flow on…

This is the second paper in a series of works devoted to nonholonomic Ricci flows. By imposing non-integrable (nonholonomic) constraints on the Ricci flows of Riemannian metrics we can model mutual transforms of generalized Finsler-Lagrange…

Consider a Riemannian manifold $(M^{m}, g)$ whose volume is the same as the standard sphere $(S^{m}, g_{round})$. If $p>\frac{m}{2}$ and $\int_{M} \left\{ Rc-(m-1)g\right\}_{-}^{p} dv$ is sufficiently small, we show that the normalized…

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse…

We present a manifold-based machine learning encoder-decoder method for learning dynamics in time, notably partial differential equations (PDEs), in which the manifold latent space evolves according to Ricci flow. This can be accomplished…

We give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type), exhibiting a combination of expanding and static behavior.

We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along…