There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional algebra with Cartan operator ∂/∂t. Likewise, the Calabi flow arises as Toda field equation associa...

Let M be a compact n-dimensional manifold, n ≥ 2, with metric g(t) evolving by the Ricci flow ∂gij/∂t = −2Rij in (0, T ) for some T ∈ R + ∪ {∞} with g(0) = g0. Let λ0(g0) be the first eigenvalue of the operator −∆g0 + R(g0) 4 with respect to g0. We extend a recent result of R. Ye and prove uniform logarithmic Sobolev inequality and uniform Sobolev inequalities along the Ricci flow for any n ≥ 2...

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 solution of the conjugate heat equation as k → ∞. We will prove the uniqueness of the f...

which is sharp as indicated in the Euclidean case. However even if M contains a small compact region where the Ricci curvature is not nonnegative, estimate (1.1) becomes very much different from (1.2) when r is large, due to the presence of the √ k term. Whether estimate (1.2) is stable under perturbation has been an open question for some time, in light of the known stability results on weaker...

The Lichnerowicz conjecture asserted that every harmonic Riemannian manifold is locally isometric to a two-point homogeneous space. In 1992, E. Damek and F. Ricci produced a family of counter-examples to this conjecture, which arise as abelian extensions of two-step nilpotent groups of type-H. In this paper we consider a broader class of Riemannian manifolds: solvmanifolds of Iwasawa type with ...

As a consequence of the Bochner formula for Bismut connection acting on gradients, we show sharp universal Poincaré and log-Sobolev inequalities along solutions to generalized Ricci flow. Using two-form potential define twisted spacetime which determines an adapted Brownian motion frame bundle, yielding Malliavin gradient path space. We this operator, leading characterizations flow in terms typ...

1 Introduction The uniformalization of Kähler manifold has been an important topic in geometry for long. The famous Frankel Conjecture states: Compact n-dimensional Kähler Manifolds of Positive Bisectional Curvature are Bi-holomorphic to P n C. [10] by using stable harmonic maps in the context of Kähler geometry. Actually, Mori proved the Harthshorne Conjecture: Every irreducible n-dimensional ...