Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Simple closed geodesics on most Alexandrov surfaces

We study the existence of simple closed geodesics on most (in the sense of Baire category) Alexandrov surfaces with curvature bounded below, compact and without boundary. We show that it depends on both the curvature bound and the topology of the surfaces.     

Boundary value problems for rotationally symmetric mean curvature flows

The motion of surfaces by their mean curvature has been studied by several authors from different points of view. K. A. Brake studied this problem from the geometric measure theory point of view, the parametric problem was studied by G. Huisken [5]. Nonparametric mean curavture flow with boundary conditions was studied in [6] and [7]. Rotationally symmetric mean curvature flows have been treated by G. Dziuk, B. Kawohl [3], but also by S. Altschuler, S. B. Angenent and Y. Giga [2]. In this paper we consider the case in which the initial surface has rotational symmetry and we shall generalize the results in [3] in the sense that we shall give more general boundary conditions which enforce the formation of a singularity in finite time. The proofs rely entirely on parabolic maximum principles. Received: 6 September 2006     

The convex hull property and topology of hypersurfaces with nonnegative curvature

We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certain required conditions. This gives a convex hull property, dual to the classical one for surfaces with nonpositive curvature. A version of this result in the nonsmooth category is obtained as well. We show that our boundary conditions determine the topology of the surface up to at most two choices. The proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing procedure.     

The Ricci flow in a class of solvmanifolds

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the ω-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.     

Ricci Yang-Mills flow on surfaces

We study the behavior of the Ricci Yang-Mills flow for U(1) bundles on surfaces. By exploiting a coupling of the Liouville and Yang-Mills energies we show that existence for the flow reduces to a bound on the isoperimetric constant or the L⁴ norm of the bundle curvature. We furthermore completely describe the behavior of long time solutions of this flow on surfaces. Finally, in Appendix A we classify all gradient solitons of this flow on surfaces.     

Curvature estimates for the Ricci flow II

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kähler-Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of this norm. These results also serve as characterization of blow-up singularities.     

Zur Interpretation der Gaußschen Osterformel und ihrer Ausnahmeregeln

In this paper, I present a revised version of Gauss's Easter formula, which is clearer than the original Easter formula and in which certain exceptions are eliminated. I also describe a method for proving calendar algorithms.Copyright 1997 Academic Press.Die Gaußsche Osterformel wird von einem internen Fehler befreit. Dadurch können die viel kritisierten Ausnahmeregeln entfallen. Es wird eine umgebaute Osterformel angegeben, die besser lesbar und verstehbar ist als die ursprüngliche Gaußsche Osterformel. Es wird eine Beweismethode für Kalenderalgorithmen mitgeteilt.Copyright 1997 Academic Press.È stata rielaborata la formula di Gauss per il calcolo della data pasquale così da rendere superflue le eccezioni. È stato proposto un nuovo concetto di tale formula che risulta meglio leggibile ed intellegibile dell'originale formula di Gauss. È stato descritto un metodo per provare gli algoritmi del calendario.     

Mean curvature flow with free boundary – Type 2 singularities

In this paper we give sufficient conditions that guarantee the mean curvature flow with free boundary on a pinched cylinder develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.     

Existence of constant mean curvature graphs in hyperbolic space

We give an existence result for constant mean curvature graphs in hyperbolic space . Let be a compact domain of a horosphere in whose boundary is mean convex, that is, its mean curvature (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that , then there exists a graph over with constant mean curvature H and boundary . Umbilical examples, when is a sphere, show that our hypothesis on H is the best possible. Received July 18, 1997 / Accepted April 24, 1998     

The condition on the stability of stationary lines in a curvature flow in the whole plane

The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O(t−1/2) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane.     

Flot de courbure moyenne modifiée avec obstacle conique

A rotationally symmetric, compact, oriented, connected, uniformly convex hypersurface M₀ of , with boundary ∂M₀ in a rotationally symmetric cone S, is evolving under volume-preserving mean curvature flow. Then for n?2, we obtain gradient and curvature estimates, leading to long-time existence of the flow, and convergence to a part of a round sphere.     

On the Dirichlet problem for the prescribed mean curvature equation over general domains

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in Rn+1 over general domains Ω without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.     

Curvature estimates for the Ricci flow I

In this paper we present several curvature estimates for solutions of the Ricci flow and the modified Ricci flow (including the volume normalized Ricci flow and the normalized Kähler-Ricci flow), which depend on the smallness of certain local \(L^{\frac{n}{2}}\) integrals of the norm of the Riemann curvature tensor |Rm|, where n denotes the dimension of themanifold. These local integrals are scaling invariant and very natural.     

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.     

The self-similar expanding curve for the curvature flow equation

We study a two-point free boundary problem for the curvature flow equation. By studying the corresponding nonlinear initial value problem, we obtain the existence and uniqueness of the forward self-similar solution of this problem. The corresponding curve is called the self-similar expanding curve. We also derive the asymptotic stability of this curve.

     

The Webster scalar curvature flow on CR sphere. Part II

This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f. Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f.     

Bounds for the first Dirichlet eigenvalue attained at an infinite family of Riemannian manifolds

LetM be a compact Riemannian manifold with smooth boundary M. We get bounds for the first eigenvalue of the Dirichlet eigenvalue problem onM in terms of bounds of the sectional curvature ofM and the normal curvatures of M. We discuss the equality, which is attained precisely on certain model spaces defined by J. H. Eschenburg. We also get analog results for Kähler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kähler manifold andP being a compact real hypersurface ofM.Work partially supported by a DGICYT Grant No. PB94-0972 and by the E.C. Contract CHRX-CT92-0050 GADGET II.     

Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form , where and F is a positive, strictly monotone and 1‐homogeneous curvature function. In particular this class includes the mean curvature . We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews–McCoy–Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.