Searcharxiv⌕ Search

arXiv subjects

S. Brendle

Publications and source records attributed to S. Brendle.

66 records · Page 4Linked to original sources
arXiv2009

A boundary value problem for minimal Lagrangian graphs

Let Ωand \tildeΩ be uniformly convex domains in \mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: Ω\to \tildeΩ such that the graph Σ= \{(x,f(x)): x \in Ω\} is a minimal Lagrangian submanifold of \mathbb{R}^n \times \mathbb{R}^n.

math.AP↗
arXiv2009

An existence theorem for the Yamabe problem on manifolds with boundary

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.

math.DG↗
arXiv2009

Sphere Theorems in Geometry

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 here new results concerning complete manifolds with pinched curvature.

math.DG↗
arXiv2009

Blow-up phenomena for the Yamabe equation

Let (M,g) be a compact Riemannian manifold of dimension n \geq 3. The Compactness Conjecture asserts that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M,g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dimensions n \geq 52.

math.DG↗
arXiv2009

Blow-up phenomena for the Yamabe equation II

Let n be an integer such that 25 \leq n \leq 51. We construct a smooth metric g on S^n with the property that the set of constant scalar curvature metrics in the conformal class of g is not compact.

math.DG↗
arXiv2008

A general convergence result for the Ricci flow in higher dimensions

Let (M,g_0) be a compact Riemannian manifold of dimension n \geq 4. We show that the normalized Ricci flow deforms g_0 to a constant curvature metric provided that (M,g_0) x R has positive isotropic curvature. This condition is stronger than 2-positive flag curvature but weaker than 2-positive curvature operator.

math.DG↗
arXiv2008

Manifolds with 1/4-pinched Curvature are Space Forms

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.

math.DG↗
arXiv2008

Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane

Let N be a complete, simply-connected surface of constant curvature κ\leq 0. Moreover, suppose that Ωand \tildeΩ are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from Ωto \tildeΩ whose graph is a minimal submanifold of N \times N.

math.DG↗