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Register a new userThe gradient flow of the $L^2$ curvature energy near the round sphere
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Authors
Jeff Streets
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We investigate the low-energy behavior of the gradient flow of the $L^2$ norm of the Riemannian curvature on four-manifolds. Specifically, we show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive Yamabe constant and small initial energy.
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We show some results for the $L^2$ curvature flow linked by the theme of addressing collapsing phenomena. First we show long time existence and convergence of the flow for $SO(3)$-invariant initial data on $S^3$, as well as a long time existence and convergence statement for three-manifolds with initial $L^2$ norm of curvature chosen small with respect only to the diameter and volume, which are both necessary dependencies for a result of this kind. In the critical dimension $n = 4$ we show a related low-energy convergence statement with an additional hypothesis. Finally we exhibit some finite time singularities in dimension $n geq 5$, and show examples of finite time singularities in dimension $n geq 6$ which are collapsed on the scale of curvature.
We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $sigma in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math. 187, no. 2, 395-506, 2012) for the scalar curvature flow on the standard unit sphere.
We prove that for a solution $(M^n,g(t))$, $tin[0,T)$, where $T<infty$, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant $C$ on $M^ntimes [0,T)$, the curvature tensor stays uniformly bounded on $M^ntimes [0,T)$. Some other results are also presented.
We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $-1$ can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $-C$. Here the time of existence and the constant $C$ only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kaehler case. We also get a local version of the main theorem. As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds).
This is a contribution to the program of dynamical approach to mean curvature flow initiated by Colding and Minicozzi. In this paper, we prove two main theorems. The first one is local in nature and the second one is global. In this first result, we pursue the stream of ideas of cite{CM3} and get a slight refinement of their results. We apply the invariant manifold theory from hyperbolic dynamics to study the dynamics close to a closed shrinker that is not a sphere. In the second theorem, we show that if a hypersurface under the rescaled mean curvature flow converges to a closed shrinker that is not a sphere, then a generic perturbation on initial data would make the flow leave a small neighborhood of the shrinker and never come back.
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