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Li-Yau Harnack Estimates for a Heat-Type Equation Under the Geometric Flow Li, Yi ; in Potential Analysis (2018) Detailed reference viewed: 143 (12 UL)Harnack estimates for a nonlinear parabolic equation under Ricci flow Li, Yi ; in Differential Geometry & its Applications (2018), 56 In this paper, we consider the Harnack estimates for a nonlinear parabolic equation under the Ricci flow. The gradient estimates for positive solutions as well as Li-Yau type inequalities are also given. Detailed reference viewed: 216 (22 UL)Local curvature estimates for the Ricci-harmonic flow Li, Yi in Math ArXiv (2018) In this paper we give an explicit bound of Δ_g(t)u(t)and the local curvature estimates for the Ricci-harmonic flow under the condition that the Ricci curvature is bounded along the flow. In the second ... [more ▼] In this paper we give an explicit bound of Δ_g(t)u(t)and the local curvature estimates for the Ricci-harmonic flow under the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature estimates are extended to a class of generalized Ricci flow, introduced by the author \cite{LY1}, whose stable points give Ricci-flat metrics on a complete manifold, and which is very close to the (K,N)-super Ricci flow recently defined by Xiangdong Li and Songzi Li \cite{LL2014}. Next we propose a conjecture for Einstein's scalar field equations motivated by a result in the first part and the bounded L^2-curvature conjecture recently solved by Klainerman, Rodnianski and Szeftel \cite{KRS2015}. In the last two parts of this paper, we discuss two notions of "Riemann curvature tensor" in the sense of Wylie-Yeroshkin \cite{KW2017, KWY2017, Wylie2015, WY2016}, respectively, and Li \cite{LY3}, whose "Ricci curvature" both give the standard Bakey-\'Emery Ricci curvature \cite{BE1985}, and the forward and backward uniqueness for the Ricci-harmonic flow. [less ▲] Detailed reference viewed: 65 (3 UL)Long time existence and bounded scalar curvature in the Ricci-harmonic flow Li, Yi in Journal of Differential Equations (2018), 265(1), 69-97 In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we ... [more ▼] In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the ``Riemann curvature'' for the Ricci-harmonic flow generalizing a recently result of Simon \cite{Simon2015}. [less ▲] Detailed reference viewed: 182 (17 UL)Local curvature estimates for the Laplacian flow Li, Yi E-print/Working paper (2018) Detailed reference viewed: 90 (8 UL)Harnack estimates for nonlinear parabolic equations under the Ricci flow Li, Yi ; E-print/Working paper (2017) In this paper, we consider first the Li-Yau Harnack estimates for a nonlinear parabolic equation $\partial_{t}u=\Delta_{t}u-qu -au(\ln u)^{\alpha}$ under the Ricci flow, where $\alpha>0$ is a constant. To ... [more ▼] In this paper, we consider first the Li-Yau Harnack estimates for a nonlinear parabolic equation $\partial_{t}u=\Delta_{t}u-qu -au(\ln u)^{\alpha}$ under the Ricci flow, where $\alpha>0$ is a constant. To extend these estimates to a more general situation, in the second part, we consider the gradient estimates for a positive solution of the nonlinear parabolic equation $\partial _{t}u=\Delta _{t}u+hu^{p}$ on a Riemannian manifold whose metrics evolve under the geometric flow $\partial _{t}g(t)=-2S_{g(t)}$. To obtain these estimates, we introduce a quantity $\underline{\boldsymbol{S}}$ along the flow which measures whether the tensor $S_{ij}$ satisfies the second contracted Bianchi identity. Under conditions on ${\rm Ric}_{g(t)}, S_{g(t)}$, and $\underline{\boldsymbol{S}}$, we obtain the gradient estimates. [less ▲] Detailed reference viewed: 47 (4 UL)Generalized Ricci flow II: Existence for complete noncompact manifolds Li, Yi E-print/Working paper (2017) Detailed reference viewed: 209 (18 UL)Long time existence of Ricci-harmonic flow Li, Yi in Frontiers of Mathematics in China (2016), 11(5), 1313-1334 We give a survey about recent results on Ricci-harmonic flow Detailed reference viewed: 120 (14 UL)Harnack estimates for a heat-type equation under the Ricci flow Li, Yi ; in Journal of Differential Equations (2016), 260(4), 3270-3301 In this paper, we consider the gradient estimates for a positive solution of the nonlinear parabolic equation ∂tu=Δtu+hup∂tu=Δtu+hup on a Riemannian manifold whose metrics evolve under the Ricci flow. Two ... [more ▼] In this paper, we consider the gradient estimates for a positive solution of the nonlinear parabolic equation ∂tu=Δtu+hup∂tu=Δtu+hup on a Riemannian manifold whose metrics evolve under the Ricci flow. Two Harnack inequalities and other interesting results are obtained. [less ▲] Detailed reference viewed: 135 (12 UL)A geometric heat flow for vector fields Li, Yi ; in Science China Mathematics (2015), 58(4), 673-688 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 ... [more ▼] 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 to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on Kähler manifolds will be studied by Li and Liu [less ▲] Detailed reference viewed: 81 (4 UL)Li-Yau-Hamilton estimates and Bakry-Emery-Ricci curvature Li, Yi in Nonlinear Analysis: Theory, Methods & Applications (2015), 113 In this paper we derive Cheng–Yau, Li–Yau, Hamilton estimates for Riemannian manifolds with Bakry–Emery–Ricci curvature bounded from below, and also global and local upper bounds, in terms of ... [more ▼] In this paper we derive Cheng–Yau, Li–Yau, Hamilton estimates for Riemannian manifolds with Bakry–Emery–Ricci curvature bounded from below, and also global and local upper bounds, in terms of Bakry–Emery–Ricci curvature, for the Hessian of positive and bounded solutions of the weighted heat equation on a closed Riemannian manifold. [less ▲] Detailed reference viewed: 110 (7 UL)On an extension of the H^k mean curvature flow of closed convex hypersurfaces Li, Yi in Geometriae Dedicata (2014), 172(1), 147-154 Detailed reference viewed: 38 (2 UL)Mabuchi and Aubin-Yau functionals over complex surfaces Li, Yi in Journal of Mathematical Analysis and Applications (2014), 416(1), 81-98 Detailed reference viewed: 71 (1 UL)A priori estimates for Donaldson's equation over compact Hermitian manifolds Li, Yi in Calculus of Variations & Partial Differential Equations (2014), 50(3-4), 867-882 Detailed reference viewed: 63 (3 UL)Eigenvalues and entropies under the harmonic-Ricci flow Li, Yi in Pacific Journal of Mathematics (2014), 267(1), 141-184 Detailed reference viewed: 64 (4 UL)Li-Yau estimates for a nonlinear parabolic equation on manifolds Li, Yi ; in Mathematical Physics, Analysis and Geometry (2014) Detailed reference viewed: 70 (2 UL)On an extension of the H^k n curvature flow Li, Yi in Science China Mathematics (2012), 55(1), 99-118 Detailed reference viewed: 29 (4 UL)Generalized Ricci flow I: Higher-derivative estimates for compact manifolds Li, Yi in Analysis & PDE (2012), 5(4), 747-775 Detailed reference viewed: 83 (6 UL)Harnack inequality for the negative power Gaussian curvature flow Li, Yi in Proceedings of the American Mathematical Society (2011), 139(10), 3070-3717 Detailed reference viewed: 92 (3 UL)Mabuchi and Aubin-Yau functionals over complex three-folds Li, Yi E-print/Working paper (2010) Detailed reference viewed: 38 (1 UL) |
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