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Martingales on manifolds with time-dependent connection Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Journal of Theoretical Probability (2015), 28(3), 1038-1062 We define martingales on manifolds with time-dependent connection, extending in this way the theory of stochastic processes on manifolds with time-changing geometry initiated by Arnaudon, Coulibaly and ... [more ▼] We define martingales on manifolds with time-dependent connection, extending in this way the theory of stochastic processes on manifolds with time-changing geometry initiated by Arnaudon, Coulibaly and Thalmaier (2008). We show that some, but not all properties of martingales on manifolds with a fixed connection extend to this more general setting. [less ▲] Detailed reference viewed: 306 (45 UL)An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Potential Analysis (2015), 42(2), 483-497 We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is ... [more ▼] We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed metrics with nonnegative Ricci curvature). As applications, we classify nonnegative ancient solutions to the heat equation according to their entropies. In particular, we show that a nonnegative ancient solution whose entropy grows sublinearly on a manifold evolving under super Ricci flow must be constant. The assumption is sharp in the sense that there do exist nonconstant positive eternal solutions whose entropies grow exactly linearly in time. Some other results are also obtained. [less ▲] Detailed reference viewed: 388 (38 UL)A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Stochastic Processes and their Applications (2014), 124(11), 3535-3552 We first prove stochastic representation formulae for space–time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems ... [more ▼] We first prove stochastic representation formulae for space–time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions. Space–time harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations. [less ▲] Detailed reference viewed: 238 (29 UL)Harnack estimates for geometric flows, applications to Ricci flow coupled with harmonic map flow Guo, Hongxin ; in Geometriae Dedicata (2014), 169(1), 411-418 We derive Harnack estimates for heat and conjugate heat equations in abstract geometric flows. The main results lead to new Harnack inequalities for a variety of geometric flows. In particular, Harnack ... [more ▼] We derive Harnack estimates for heat and conjugate heat equations in abstract geometric flows. The main results lead to new Harnack inequalities for a variety of geometric flows. In particular, Harnack inequalities for the Ricci flow coupled with Harmonic map flow are obtained. [less ▲] Detailed reference viewed: 82 (2 UL)A note on Chow's entropy functional for the Gauss curvature flow Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Comptes Rendus de l'Académie des Sciences. Série I. Mathématique (2013), 351(21-22), 833-835 Based on the entropy formula for the Gauss curvature flow introduced by Bennett Chow, we define an entropy functional which is monotone along the unnormalized flow and whose critical point is a shrinking ... [more ▼] Based on the entropy formula for the Gauss curvature flow introduced by Bennett Chow, we define an entropy functional which is monotone along the unnormalized flow and whose critical point is a shrinking self-similar solution. [less ▲] Detailed reference viewed: 222 (13 UL)An entropy formula relating Hamiltonʼs surface entropy and Perelmanʼs W entropy Guo, Hongxin in Comptes Rendus. Mathématique (2013), 351(3-4), 115-118 In this note, based on Hamiltonʼs surface entropy formula, we construct an entropy formula of Perelmanʼs type for the Ricci flow on a closed surface with positive curvature. Similar to Perelmanʼs WW ... [more ▼] In this note, based on Hamiltonʼs surface entropy formula, we construct an entropy formula of Perelmanʼs type for the Ricci flow on a closed surface with positive curvature. Similar to Perelmanʼs WW entropy, the critical point of our entropy is the gradient shrinking soliton; however, there is no conjugate heat equation involved. This shows a close relation between Hamiltonʼs entropy and Perelmanʼs W entropy on closed surfaces. [less ▲] Detailed reference viewed: 40 (2 UL)Entropy and lowest eigenvalue on evolving manifolds Guo, Hongxin ; Philipowski, Robert ; Thalmaier, Anton in Pacific J. Math. (2013), 264(1), 61-81 Detailed reference viewed: 224 (21 UL) |
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