Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.

Manifolds with 1/4-pinched Curvature are Space Forms

In 1982, R Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincare conjecture Furthermore, various convergence theorems have been established This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow The proofs rely mostly on maximum principle arguments Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold whose sectional curvatures all lie in the interval (1,4] is diffeomorphic to a spherical space form This question has a long history, dating back to a seminal paper by H E Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen This text originated from graduate courses given at ETH Zurich and Stanford University, and is directed at graduate students and researchers The reader is assumed to be familiar with basic Riemannian geometry, but no previous knowledge of Ricci flow is required

Ricci Flow and the Sphere Theorem

In 1982, Hamilton [41] introduced the Ricci flow to study compact three-manifolds with positive Ricci curvature. Through decades of works of many mathematicians, the Ricci flow has been widely used to study the topology, geometry and complex structure of manifolds. In particular, Hamilton’s fundamental works (cf. [12]) in the past two decades and the recent breakthroughs of Perelman [80, 81, 82] have made the Ricci flow one of the most intricate and powerful tools in geometric analysis, and led to the resolutions of the famous Poincare conjecture and Thurston’s geometrization conjecture in three-dimensional topology. In this survey, we will review the recent developments on the Ricci flow and give an outline of the Hamilton-Perelman proof of the Poincare conjecture, as well as that of a proof of Thurston’s geometrization conjecture.

https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2007/0012/0001/SDG-2007-0012-0001-a003.pdf

Recent developments on the Hamilton’s Ricci Flow

Let (M,g) be an Einstein manifold of dimension n≥4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric

Einstein manifolds with nonnegative isotropic curvature are locally symmetric

In this paper, we study the Ricci flow on higher dimensional compact manifolds. We prove that nonnegative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to four. In order to do so, we introduce a new technique to prove that curvature functions defined on the orthonormal frame bundle are preserved by the Ricci flow. At a minimum of such a function, we compute the first and second derivatives in the frame bundle. Using an algebraic construction, we can use these expressions to show that the nonlinearity is positive at a minimum. Finally, using the maximum principle, we can show that the Ricci flow preserves the cone of curvature operators with nonnegative isotropic curvature.

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Isotropic Curvature and the Ricci Flow

The Ricci flow on a compact four-manifold preserves the condition of pointwise 1/4- pinching of flag curvatures. Any compact Riemannian four-manifold with 1/4-pinched flag curvatures is either isometric to CP 2 or diffeomorphic to a space-form.

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/2009/0013/0002/AJM-2009-0013-0002-a005.pdf

Four-manifolds with 1/4-pinched Flag Curvatures

In the paper Muller–Sverak (J Differ Geom 42(2):229–258, 1995) conformally immersed surfaces with finite total curvature were studied. In particular it was shown that surfaces with total curvature A28 in dimension three were embedded and conformal to the plane with one end. Here, using techniques from Kuwert–Li (W 2,2-conformal immersions of a closed Riemann surface into R n . arXiv:1007.3967v2 [math.DG], 2010), we will show that if the total curvature A28 , then we are either embedded and conformal to the plane, isometric to a catenoid or isometric to Enneper’s minimal surface. In fact the technique of our proof shows that if we are conformal to the plane, then if n ≥ 3 and A216 then Σ is embedded or Σ is the image of a generalized catenoid inverted at a point on the catenoid. In order to prove these theorems, we prove a Gauss–Bonnet theorem for surfaces with complete ends and isolated finite area singularities which extends a theorem of Jorge-Meeks (Topology 22(2):203–221, 1983). Using this theorem, we then prove an inversion formula for the Willmore energy.

Geometric rigidity for analytic estimates of Müller–Šverák

We study mean curvature flow in the sphere with the quadratic curvature condition |A| ≤ 1 n−2H 2 + 4K which is related but different to two-convexity for submanifolds of the sphere. We classify type I singularities with no further hypotheses. If H > 0 then we apply the Huisken-Sinestrari convexity estimates to this situation and show that we can classify type II singularities. This shows that at a singularity the surface is asymptotically convex. We then prove cylindrical estimates for the mean curvature flow and a pointwise gradient estimate which shows that near a singularity the surface is quantitatively convex.

/pdf/convexity-and-cylindrical-estimates-for-mean-curvature-flow-1f5m8svk0g.pdf

Convexity and cylindrical estimates for mean curvature flow in the sphere

We analyse sequences of discs conformally immersed in $$ \mathbb{R }^ n$$ with energy $$ \int _{ D} |A_k |^ 2 \le \gamma _n$$, where $$ \gamma _n = 8\pi $$ if $$ n=3$$ and $$ \gamma _n = 4 \pi $$ when $$n\ge 4$$. We show that if such sequences do not weakly converge to a conformal immersion, then by a sequence of dilations we obtain a complete minimal surface with bounded total curvature, either Enneper’s minimal surface if $$ n=3$$ or Chen’s minimal graph if $$ n \ge 4$$. In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313–340, 2012; Rivière, Adv Calculus Variations 6(1), 1–31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then $$\liminf _ {k\rightarrow \infty } \mathcal W ( f_k )\ge 8\pi $$. We apply the above analysis to show that in $$ \mathbb{R }^3$$ if the sequence diverges so that $$ \lim _{ k \rightarrow \infty } \mathcal W (f_k) =8\pi $$ then there exists a sequence of Möbius transforms $$ \sigma _{k}$$ such that $$ \sigma _k\circ f _k$$ converges weakly to a catenoid.

H. T. Nguyen

Papers

Isotropic Curvature and the Ricci Flow

Four-manifolds with 1/4-pinched Flag Curvatures

Geometric rigidity for analytic estimates of Müller–Šverák

Convexity and cylindrical estimates for mean curvature flow in the sphere

Geometric rigidity for sequences of $$ W^{2,2}$$ conformal immersions