Let (M,g) be a compact Riemannian manifold of dimension 3, and let F denote the collection of all embedded incompressible surfaces homeomorphic to RP^2. We study the infimum of the areas of all surfaces in F. This quantity makes sense whenever F is non-empty, and is related to the systole of (M,g). Our main result is an upper bound for this quantity in terms of the minimum of the scalar curvature of (M,g). This inequality is sharp on RP^3 equipped with its standard metric. Conversely, if equality holds, then (M,g) is isometric to a spherical space form. The proof uses the formula for the second variation of area, and Hamilton's Ricci flow.
Let (M,g) be a compact Riemannian manifold with boundary. We consider the problem (first studied by Escobar in 1992) of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary. Using a local test function construction, we are able to settle most cases left open by Escobar's work. Moreover, we reduce the remaining cases to the positive mass theorem.
In this note, we describe an interpolation inequality in the setting of Ricci flow and L-distance. This inequality is motivated by a classical inequality of Prekopa and Leindler, and is related to Perelman's monotonicity formula for the reduced volume. The proof relies on Topping's theory of L-optimal transport.
In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton's Ricci flow. We also obtain new results on complete manifolds with pinched curvature. These results generalize a theorem of Chen and Zhu. Finally, we prove a four-dimensional version of Fraser's theorem concerning the fundamental groups of manifolds with positive isotropic curvature.
Consider two uniformly convex domains in R^n with smooth boundary. We show that there exists a diffeomorphism f from the first domain to the second one such that the graph of f is a minimal Lagrangian submanifold of R^n x R^n.
The existence of minimal Lagrangian graphs turns out to be equivalent to the solvability of a certain fully nonlinear partial differential equation. In fact, the graph of f is Lagrangian if and only if f is the gradient of a scalar function u. Furthermore, the requirement that the graph of f is minimal Lagrangian can be reduced to a fully nonlinear partial differential equation for the function u. The main result of the paper asserts that, given two uniformly convex domains in R^n with smooth boundary, there exists a smooth, convex solution u of this partial differential equation with the property that the gradient of u maps the first domain to the second one. Furthermore, the function u is unique up to addition of constants. This type of problem is commonly referred to as the "second" boundary value problem (the "first" boundary value problem being the Dirichlet problem). The solvability of the Dirichlet boundary value problem for this partial differential equation was established in earlier work of Caffarelli, Nirenberg, and Spruck.
In this paper, we show that an Einstein manifold with positive isotropic curvature necessarily has constant sectional curvature. Moreover, we show than an Einstein manifold with nonnegative isotropic curvature must be locally symmetric. This generalizes results of Berger and Tachibana. To prove the result, we study the minimum isotropic curvature and apply the maximum principle.
In 1993, R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. This estimate plays a central role in the analysis of singularities to the Ricci flow. It is an important question whether some version of Hamilton's Harnack inequality holds when M fails to have nonnegative curvature operator. In this paper, we show that Hamilton's differential Harnack inequality holds under the weaker condition that M x R^2 has nonnegative isotropic curvature. The latter condition is weaker than nonnegative curvature operator and stronger than nonnegative sectional curvature.
Let (M,g_0) be a compact Riemannian manifold of dimension at least 4. We show that the Ricci flow evolves g_0 to a constant curvature metric provided that (M,g_0) x R has positive isotropic curvature. This curvature condition is stronger than 2-positive flag curvature but weaker than 2-positive curvature operator. When M is three-dimensional, the condition that (M,g_0) x R has positive isotropic curvature is equivalent to the condition that (M,g_0) has positive Ricci curvature. Hence, the main theorem of this paper generalizes a theorem of Hamilton.
The proof of this theorem relies on the following result, which is of interest in itself: if (M,g_0) x S^2(1) has nonnegative isotropic curvature, then this remains so if the metric evolves under the Ricci flow. Here, S^2(1) denotes a two-dimensional sphere of radius 1. The key point is that the radius of the two-sphere is constant in time.
In this paper, we study the borderline case in the Differentiable Sphere Theorem. We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we give a classification of all compact, locally irreducible Riemannian manifolds M with the property that M x R^2 has non-negative isotropic curvature.
The proof relies on a variant of J.M. Bony's strict maximum principle for degenerate elliptic equations. The techniques developed in this paper actually work in a much more general setting. They have since found applications to other borderline situations in Ricci flow, including the generalized Frankel conjecture. The results in this paper also play a role in Petersen and Tao's classification of almost 1/4-pinched manifolds.
Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow evolves g_0 to a constant curvature metric. In particular, M is diffeomorphic to a spherical space form. This confirms the Differentiable Sphere Theorem (conjectured by Rauch in 1951).
The proof employs Hamilton's Ricci flow. We first show that nonnegative isotropic curvature is preserved by the Ricci flow in all dimensions. The proof of this result relies the maximum principle, and exploits subtle algebraic inequalities. We then consider the condition that M x R^2 has nonnegative isotropic curvature. This condition is again preserved by the Ricci flow, and plays a key role in the argument.
Consider two domains in the hyperbolic plane H^2 with smooth boundary. Suppose that both domains are uniformly convex and have the same area with respect to the hyperbolic metric. We show that there exists a diffeomorphism f from one of these domains to the other such that the graph of f is a minimal Lagrangian surface in H^2 x H^2. Moreover, the set of all such diffeomorphisms can be parametrized by the circle S^1.
This is a survey article on topics related to the Yamabe problem. We describe various compactness and non-compactness results for the Yamabe equation. We also discuss the asymptotic behavior of the parabolic Yamabe flow.
Let n be an integer between 25 and 51. In this paper, we construct a smooth metric g on S^n such that the set of solutions to the Yamabe equation on (S^n,g) is non-compact. This result is sharp: if (M,g) is a compact spin manifold of dimension less than 25, then the set of solutions to the Yamabe equation is known to be compact.
In this paper, we construct counterexamples to R. Schoen's Compactness Conjecture for the Yamabe equation. More precisely, for each integer n > 51, we construct a smooth metric g on S^n such that the set of solutions to the Yamabe equation on (S^n,g) fails to be compact. The blow-up sequences have increasing energy. In fact, for each of these blow-up sequences, the energy converges from below to the energy of the round metric on S^n.
Let (M,g_0) be a compact Riemannian manifold of dimension at least 6. Moreover, suppose that there is no point on M where the Weyl tensor vanishes to an order greater than [(n-6)/2]. Under this condition, we are able to construct a family of test functions with Yamabe energy less than Y(S^n). As a corollary, we show that the Yamabe flow with initial metric g_0 converges to a metric of constant scalar curvature as t tends to infinity.
We also show that the assumption on the Weyl tensor can be dropped if the positive mass theorem holds. Since the positive mass theorem is known for spin manifolds, we conclude that the Yamabe flow always converges to a constant scalar curvature metric if M is spin.
Let (M,g_0) be a compact Riemannian manifold, and let g(t) be the solution of the Yamabe flow with initial metric g_0. R. Hamilton has conjectured that the metrics g(t) converge to a metric of constant scalar curvature as t tends to infinity. In this paper, we confirm Hamilton's conjecture, assuming that there exists a suitable family of test functions with Yamabe energy less than Y(S^n). If the dimension of M is less than 6, the existence of such test functions follows from R. Schoen's work on the Yamabe problem. The same argument works for locally conformally flat manifolds of any dimension. As a consequence, we are able to verify Hamilton's conjecture in these cases.