Bounded Mean Curvature Research Articles

Maximum principles at infinity for surfaces of bounded mean curvature in R^3 and H^3

Let M 1 , M 2 be disjoint surfaces in R 3 or H 3 with (possibly empty) boundaries ∂M 1 , ∂M 2 and bounded mean curvature. We establish a maximum principle at infinity for these surfaces by proving that under certain conditions on their curvatures, M 1 and M 2 cannot approach each other asymptotically.

Higher order energy expansions for some singularly perturbed Neumann problems

We consider the following singularly perturbed semilinear elliptic problem: ε 2 Δu−u+u p=0 in Ω, u>0 in Ω and ∂u ∂ν =0 on ∂Ω, where Ω is a bounded smooth domain in R N , ε>0 is a small constant and p is a subcritical exponent. Let J ε[u]:=∫ Ω( ε 2 2 |∇u| 2+ 1 2 u 2− 1 p+1 u p+1) dx be its energy functional, where u∈H 1(Ω) . Ni and Takagi proved that for a single boundary spike solution u ε , the following asymptotic expansion holds J ε[u ε]=ε N 1 2 I[w]−c 1εH(P ε)+ o(ε) , where c 1>0 is a generic constant, P ε is the unique local maximum point of u ε and H( P ε ) is the boundary mean curvature function. In this Note, we obtain the following higher order expansion of J ε [ u ε ]: J ε[u ε]=ε N 1 2 I[w]−c 1εH(P ε)+ε 2[c 2(H(P ε)) 2+c 3R(P ε)]+ o(ε 2) , where c 2, c 3 are generic constants and R( P ε ) is the Ricci scalar curvature at P ε . In particular c 3>0. Applications of this expansion will be given. To cite this article: J. Wei, M. Winter, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

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.

Eigenvalue Estimates for Submanifolds with Locally Bounded Mean Curvature

We present a method to obtain lower bounds for firstDirichlet eigenvalue in terms of vector fields with positivedivergence. Applying this to the gradient of a distance functionwe obtain estimates of eigenvalue of balls inside the cut locus and of domains Ω ⊂ M ∩ BN(p, r) in submanifolds M ⊂ϕNwith locally bounded mean curvature. Forsubmanifolds of Hadamard manifolds with bounded mean curvaturethese lower bounds depend only on the dimension of the submanifold and the bound on its mean curvature.

Spectral convergence of conformally immersed surfaces with bounded mean curvature

Motivated by B6rard, Besson, and Gallot [1], we introduced in [10] and [11] a distance between compact connected Riemannian manifolds using their heat kernels and studied its basic properties. The distance is called the spectral distance SD and defined as follows: For a compact connected Riemannian manifold (M, g), we denote by PM (t, x, y) the heat kernel of the Laplace operator of M with respect to the normalized Riemannian measure IZM = dvg/Vol(M, g). Given two compact connected Riemannian manifolds M and N, a map f : M --~ N is called an s-spectral approximating map if it satisfies

Isoperimetric inequalities for parametric variational problems

We prove isoperimetric inequalities for general parametric variational double integrals F, whose Lagrangians F depend on the position vector X and on the surface normal N. As an essential tool we introduce Sauvigny's F-conformal parameters adapted to the parametric integrand and use the notion of generalized mean and Gaussian curvature. The special cases of minimal surfaces, surfaces of bounded mean curvature and ▪-minimizing surfaces are also discussed.

Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space

Let M be an n-dimensional complete non-compact submanifold in a hyperbolic space with the norm of its mean curvature vector bounded by a constant \(\alpha 0 \). In particular when M is minimal we have \(\lambda _{1}\left( M\right) \geq \frac{1}{4} \left( n-1\right)^{2} \) and this is sharp because equality holds when M is totally geodesic.

A remark on isolated singularities of surfaces with bounded mean curvature: the non-minimizing and non-perpendicular case

The length of the free boundary of twodimensional weak surfaces with bounded mean curvature is studied in the case of non-perpendicular contact angles and for non-minimizing stationary surfaces. Isolated singularities are excluded if the contact angle is bounded away from zero and if the solution is assumed to lie on one side of the supporting surface.

On the free boundary of surfaces with bounded mean curvature: the non-perpendicular case

Holder continuity up to the free boundary is proved for minimizing solutions if they meet the supporting surface in an angle which is bounded away from zero. The problem is localized by proving the continuity of the distance function, a result which is also true for stationary points.

The mean curvature and volume growth of complete noncompact submanifolds

We show that a complete noncompact submanifold with bounded mean curvature in the Euclidean or hyperbolic space has at least linear volume growth. We apply this to obtain some results on submanifolds of parallel mean curvature.

Bounded mean curvature isometric immersions of a compact Riemannian manifold with images contained in a tube

AbstractWe characterize some isometric immersions of a compact Riemannian manifold into a tube of Sn(λ) or CPn(λ) (in fact, in some more general spaces in the real case) around a totally geodesic Sq(λ) or CPq(λ) respectively, with the norm of the mean curvature of the immersion bounded from above. This bound depends on the radius of the tube, and is related with the mean curvature of its boundary.

The structure of an infinite, complete, convex hypersurface inE 4 with bounded mean curvature

The following theorem is proved. THEOREM. If on an infinite, complete, convex hypersurface F in E4 the mean curvature is 1 − ε ≤ H ≤ 1, where 0 ≤ ε ≤ 10−11, then F is a cylindrical hypersurface.

Maximum principles and nonexistence results for minimal submanifolds

We construct a quadratic form on ℝn+k of signature (n-k) which is subharmonic on any n-dimensional minimal submanifold in ℝn+k. This yields an improvement over the convex hull property of minimal submanifolds as well as necessary conditions for compact minimal submanifolds the boundaries of which lie in disconnected sets. The argument also extends to submanifolds of bounded mean curvature. Furthermore an optimal nonexistence result is derived by employing a different geometrical argument, which is based on the construction of n-dimensional catenoids.

A Geometric Maximum Principle for Surfaces of Prescribed Mean Curvature in Riemannian Manifolds

Let M be a three-dimensional Riemannian manifold and let f be some surface of prescribed mean curvature which is restricted to lie in some set J \cup S \subset M with boundary S of bounded mean curvature \mathfrak H . Assuming natural conditions, we prove that the image of f lies completely in J . An immediate consequence of this result is a sufficient condition for the existence of minimal surfaces in a set J \subset \mathbb R^3 , the boundary S of which is not \mathfrak h -convex.

Spacelike hypersurfaces with prescribed boundary values and mean curvature

We consider the boundary-value problem for the mean curvature operator in Minkowski space, and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions. Our main results hold for non-constant mean curvature, and make no assumptions about the smoothness of the boundary or boundary data.

The diameter of an immersed Riemannian manifold with bounded mean curvature

AbstractLet M be an n-dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N-dimensional (N < n) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥H∥ > H0, (H0 > 0) then R > H−10.

Immersions of bounded mean curvature: Addendum

Immersions of bounded mean curvature: Addendum

Immersions of bounded mean curvature

Immersions of bounded mean curvature

Removability of singular points on surfaces of bounded mean curvature

Removability of singular points on surfaces of bounded mean curvature

Fl�chen Beschr�nkter Mittlerer Kr�mmung in Einer Dreidimensionalen Riemannschen Mannigfaltigkeit

In recent papers HILDEBRANDT [11] and HARTH [5] proved the existence of solutions of the problem of Plateau for surfaces of bounded mean curvature with fixed and free boundaries in E3 and for minimal surfaces with free boundaries in a Riemannian manifold, respectively. Here their methods will be combined to solve the problem of Plateau for surfaces of bounded mean curvature in a Riemannian manifold. This will be done for fixed and free boundaries. Moreover, isoperimetric inequalities for the solutions will be given.