The isoperimetric problem in complete surfaces of nonnegative curvature

We show that in a complete plane with nonnegative curvature there is a perimeter minimizing set of any given area. This set is a disc whose boundary is a closed embedded curve with constant geodesic curvature.     

Some existence and nonexistence theorems for compact graphs of constant mean curvature with boundary in parallel planes

Given H≥0 and bounded convex curves α₁, ...,⇌_n, α in the plane z=0 bounding domains D₁, …, D_n, D, respectively, with if i ∈ j and with D_i ⊂ D, we obtain several results proving the existence of a constanth depending only on H and on the geometry of the curves α_i, α such that the Dirichlet problem for the constant mean curvature H equation: where may accept or not a solution.     

An L p -estimate for surfaces of constant mean curvature in {\mathbb{S}}^3

In this paper, we prove a one end theorem for complete noncompact Riemannian manifolds and apply it to complete noncompact stable minimal hypersurfaces. Received: 10 September 2007     

An exotic sphere with positive curvature almost everywhere

In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere. In 1974 Gromoll and Meyer found a metric of nonnegative sectional on an exotic 7-sphere. They showed that the metric has positive curvature at a point and asserted, without proof, that the metric has positive sectional curvature almost everywhere [4]. We will show here that this assertion is wrong. In fact, the Gromoll-Meyer sphere has zero curvatures on an open set of points. Never the less, its metric can be perturbed to one that has positive curvature almost everywhere.     

Manifolds whose curvature operator has constant eigenvalues at the basepoint

Osserman conjectured that if the curvature operatorR of a Riemannian manifoldM has constant eigenvalues, thenM is locally a rank-1 symmetric space or is flat. The pointwise question is considerably more complicated. We present examples of Riemannian manifolds so thatR has constant eigenvalues at the basepoint, butR is not the curvature operator of a rank-1 symmetric space. Research partially supported by the NSF and IHES.     

Double bubbles in<Emphasis Type="Bold">S</Emphasis><Superscript>3</Superscript> and<Emphasis Type="Bold">H</Emphasis><Superscript>3</Superscript>

We prove the double bubble conjecture in the three-sphereS ³ and hyperbolic three-spaceH ³ in the cases where we can apply Hutchings theory:

Isoscattering deformations for complete manifolds of negative curvature

We construct continuous families of nonisometric metrics on simply connected manifolds of dimension n ≥ 9which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.     

Singular integral operators on Riemann surfaces and elliptic differential systems

The derivatives of the Cauchy kernels on compact Riemann surfaces generate singular integral operators analogous to the Calderón-Zigmund operators with the kernel (t - z)² on the complex plane. These operators play an important role in studying elliptic differential equations, boundary value problems, etc. We consider here the most important case of the multi-valued Cauchy kernel with real normalization of periods. In the opposite plane case, such an operator is not unitary. Nevertheless, its norm in L₂ is equal to one. This result is used to study multi-valued solutions of elliptic differential systems.     

Proof of the double bubble curvature conjecture

An area minimizing double bubble in ℝⁿ is given by two (not necessarily connected) regions which have two prescribed n-dimensional volumes whose combined boundary has least (n−1)-dimensional area. The double bubble theorem states that such an area minimizer is necessarily given by a standard double bubble, composed of three spherical caps. This has now been proven for n = 2, 3,4, but is, for general volumes, unknown for n ≥ 5. Here, for arbitrary n, we prove a conjectured lower bound on the mean curvature of a standard double bubble. This provides an alternative line of reasoning for part of the proof of the double bubble theorem in ℝ³, as well as some new component bounds in ℝⁿ.     

Prescribing scalar and boundary mean curvature on the three dimensional half sphere

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half-three sphere, by deforming conformally its standard metric. Using blow-up analysis techniques and minimax arguments, we prove some existence and compactness results.     

Compact gradient shrinking Ricci solitons with positive curvature operator

In this article, we first derive several identities on a compact shrinking Ricci soliton. We then show that a compact gradient shrinking soliton must be Einstein, if it admits a Riemannian metric with positive curvature operator and satisfies an integral inequality. Furthermore, such a soliton must be of constant curvature.     

Local volume estimate for manifolds with<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-bounded curvature

We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite L² norm in dimension 4.     

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder in a product Riemannian manifold . It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabi on complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature. Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthday.     

Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples.     

Embedding open Riemann surfaces in Riemannian manifolds

For the Riemann surface of the topological type, we can get a conformai model in orientable Riemannian manifolds. We will prove that there is a conformally equivalent model in orientable Riemannian manifolds for a given open Riemann surface. To end up we utilize Garsia 's Continuity lemma and Brouwer's Fixed Point lemma along with the Teichmüller theory.     

On surfaces of prescribed weighted mean curvature

Utilising a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.     

Admissible convergence in Cartan-Hadamard manifolds

In the context of a complete simply connected Riemannian manifold of pinched negative curvature, we show that several families of approach domains are equivalent for convergence to points of the boundary, and for the purposes of H^p-theory.     

Small excess and Ricci curvature

It is proved that a Riemanniann-manifold with Ricci curvature ≥ (n − 1) and a lower injectivity radius bound is a sphere provided the diameter is sufficiently close to π. The author was partially supported by the NSF and the Alfred P. Sloan Foundation.     

On Hamiltonian stable minimal Lagrangian surfaces in CP2

We study Hamiltonian stable minimal Lagrangian closed submanifolds in the standard complex projective n-space CP ⁿ .It is shown that when n = 2such a surface Σis either totally geodesic or flat if the multiplicity of the Laplacian acting on C∞(Σ)is less than or equal to 6.     

Transverse curves and chain curvature in the Heisenberg group

A chain is the intersection of a complex totally geodesic subspace in complex hyperbolic 2-space with the boundary. The boundary admits a canonical contact structure, and chains are distiguished curves transverse to this structure. The space of chains is analyzed both as a quotient of the contact bundle, and as a subset of ℂP². The space of chains admits a canonical, indefinite Hermitian metric, and curves in the space of chains with null tangent vectors are shown to correspond to a path of chains tangent to a curve in the boundary transverse to the contact structure. A family of local differential chain curvature operators are introduced which exactly characterize when a transverse curve is a chain. In particular, operators that are invariant under the stabilizer of a point in the interior of complex hyperbolic space, or a point on the boundary, are developed in detail. Finally, these chain curvature operators are used to prove a generalization of Louiville's theorem: a sufficiently smooth mapping from the boundary of complex hyperbolic 2-space to itself which preserves chains must be the restriction of a global automorphism.