Area Integrand Research Articles

The Concept of Varifold

We survey - by means of 20 examples - the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural language for studying the variational theory of the area integrand if one considers, for instance, existence or regularity of stationary (or, stable) surfaces of dimension at least three, or the limiting behaviour of sequences of smooth submanifolds under area and mean curvature bounds.

Minimal surfaces in pseudohermitian geometry

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set (i.e., the set where the (p-)area integrand vanishes), we formulate some extension theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the xy-plane) in the Heisenberg group H1. In H1, identified with the Euclidean space R3, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, C2 smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard S3. This fact continues to hold when S3 is replaced by a general pseudohermitian 3-manifold. Mathematics Subject Classification (2000): 35L80 (primary); 35J70, 32V20, 53A10, 49Q10 (secondary).

Interior gradient estimates for anisotropic mean-curvature flow

In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean-curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic area integrand.

Estimates for stable hypersurfaces of prescribed F-mean curvature

We consider immersed hypersurfaces Open image in new window:Mn→ℝn+1 with prescribed anisotropic mean curvature Open image in new window. Such hypersurfaces can be characterized as critical points of parametric functionals of the type Open image in new window with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying Open image in new window. We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C3-close to the area integrand.

Integral curvature estimates for F-stable hypersurfaces

We consider immersed hypersurfaces in euclidean \(% \mathbb{R}^{{n + 1}} % \) which are stable with respect to an elliptic parametric functional with integrand F = F(N) depending on normal directions only. We prove an integral curvature estimate provided that F is sufficiently close to the area integrand, extending the classical estimate of Schoen, Simon and Yau [19] for stable minimal hypersurfaces in \(% \mathbb{R}^{{n + 1}} % \), as well as the pointwise estimate of Simon [22] for F-minimizing hypersurfaces. As a crucial point of our analysis we derive a generalized Simons inequality for the laplacian of the length of a weighted second fundamental form with respect to an abstract metric associated with F. As an application, we obtain a new Bernstein result for complete F-stable hypersurfaces of dimension \(n \leq 5\).

Pointwise curvature estimates for F-stable hypersurfaces

We consider immersed hypersurfaces in euclidean Rn+1 which are stable with respect to an elliptic parametric functional of the form F(X)=∫MF(N)dμ. We prove a pointwise curvature estimate provided that n⩽5 and F is sufficiently close to the area integrand. This extends the pointwise curvature estimates of Schoen, Simon and Yau [Acta Math. 134 (1975) 275] for stable minimal hypersurfaces in Rn+1 and of Simon [Math. Z. 154 (1977) 265] for minimizers of F. Our result follows from an integral curvature estimate and a generalized Simons inequality that were established recently [Calc. Var. Partial Differential Equations (2004), DOI: 10.1007/S00526-004-0306-5], together with a Moser type iteration argument.

A Parabolic Quasilinear Problem for Linear Growth Functionals

We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A typical example of energy functional we consider is the one given by the nonparametric area integrand f(x, \xi) = \sqrt{1 + \Vert \xi \Vert^2} , which corresponds with the time-dependent minimal surface equation. We also study the asymptotic behaviour of the solutions.

Area-Minimizing Integral Currents with Boundaries Invariant Under Polar Actions

Let $G$ be a compact, connected subgroup of $SO(n)$ acting on ${{\mathbf {R}}^n}$, and let the action of $G$ be polar. (Polar actions include the adjoint action of a Lie group $H$ on the tangent space to the symmetric space $G/H$ at the identity coset.) Let $B$ be an $(m - 1)$-dimensional submanifold without boundary, invariant under the action of $G$, and lying in the union of the principal orbits of $G$. It is shown that, if $S$ is an area-minimizing integral current with boundary $B$, then $S$ is invariant under the action of $G$. This result is extended to a larger class of boundaries, and to a class of parametric integrands including the area integrand.

Area-minimizing integral currents with boundaries invariant under polar actions

Let G G be a compact, connected subgroup of S O ( n ) SO(n) acting on R n {{\mathbf {R}}^n} , and let the action of G G be polar. (Polar actions include the adjoint action of a Lie group H H on the tangent space to the symmetric space G / H G/H at the identity coset.) Let B B be an ( m − 1 ) (m - 1) -dimensional submanifold without boundary, invariant under the action of G G , and lying in the union of the principal orbits of G G . It is shown that, if S S is an area-minimizing integral current with boundary B B , then S S is invariant under the action of G G . This result is extended to a larger class of boundaries, and to a class of parametric integrands including the area integrand.

Invariance of Solutions to Invariant Nonparametric Variational Problems

Let $f$ be a weak solution to the Euler-Lagrange equation of a convex nonparametric variational integral in a bounded open subset $D$ of ${{\mathbf {R}}^n}$. Assume the boundary $B$ of $D$ to be rectifiable. Let $D$ be a compact connected Lie group of diffeomorphisms of a neighborhood of $D \cup B$ which leave $D$ invariant and assume the variational integral to be $G$-invariant. Conditions are formulated which imply that if $f$ is continuous on $D \cup B$ and $f \circ g|B = f|B$ for $g \in G$ then $f \circ g = f$ for every $g \in G$. If the integrand $L$ is strictly convex then $f$ can be shown to have a local uniqueness property which implies invariance. In case $L$ is not strictly convex the graph ${T_f}$ of $f$ in ${{\mathbf {R}}^n} \times {\mathbf {R}}$ is interpreted as the solution to an invariant parametric variational problem, and invariance of ${T_f}$, hence of $f$, follows from previous results of the author. For this purpose a characterization is obtained of those nonparametric integrands on ${{\mathbf {R}}^n}$ which correspond to a convex positive even parametric integrand on ${{\mathbf {R}}^n} \times {\mathbf {R}}$ in the same way that the nonparametric area integrand corresponds to the parametric area integrand.

Invariance of solutions to invariant nonparametric variational problems

Let f f be a weak solution to the Euler-Lagrange equation of a convex nonparametric variational integral in a bounded open subset D D of R n {{\mathbf {R}}^n} . Assume the boundary B B of D D to be rectifiable. Let D D be a compact connected Lie group of diffeomorphisms of a neighborhood of D ∪ B D \cup B which leave D D invariant and assume the variational integral to be G G -invariant. Conditions are formulated which imply that if f f is continuous on D ∪ B D \cup B and f ∘ g | B = f | B f \circ g|B = f|B for g ∈ G g \in G then f ∘ g = f f \circ g = f for every g ∈ G g \in G . If the integrand L L is strictly convex then f f can be shown to have a local uniqueness property which implies invariance. In case L L is not strictly convex the graph T f {T_f} of f f in R n × R {{\mathbf {R}}^n} \times {\mathbf {R}} is interpreted as the solution to an invariant parametric variational problem, and invariance of T f {T_f} , hence of f f , follows from previous results of the author. For this purpose a characterization is obtained of those nonparametric integrands on R n {{\mathbf {R}}^n} which correspond to a convex positive even parametric integrand on R n × R {{\mathbf {R}}^n} \times {\mathbf {R}} in the same way that the nonparametric area integrand corresponds to the parametric area integrand.

Invariance of solutions to invariant parametric variational problems

Let G be a compact Lie group of diffeomorphisms of a connected orientable manifold M of dimension n + 1 n + 1 . Assume the orbits of highest dimension to be connected. Let Ψ \Psi be a convex positive even parametric integrand of degree n on M which is invariant under the action of G. Let T be a homologically Ψ \Psi -minimizing rectifiable current of dimension n on M, and assume there exists a G-invariant rectifiable current T ′ T’ which is homologous to T. It is shown that T is G-invariant provided Ψ \Psi satisfies a symmetry condition which makes it no less efficient for the tangent planes of T to lie along the orbits. This condition is satisfied by the area integrand in case G is a group of isometries of a Riemannian metric on M. Consequently, one obtains the corollary that if a rectifiable current T is a solution to the n-dimensional Plateau problem in M with G-invariant boundary ∂ T \partial T , and if ∂ T \partial T bounds a G-invariant rectifiable current T ′ T’ such that T − T ′ T - T’ is a boundary, then T is G-invariant. An application to the Plateau problem in S 3 {{\textbf {S}}^3} is given.

On the First Variation of a Varifold

Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our main result is a regularity theorem for weakly defined k dimensional surfaces in M whose first variation of area is summable to a power greater than k. A natural domain for any k dimensional parametric integral in M, among which the simplest is the k dimensional area integral, is the space of k dimensional varifolds in M introduced by Almgren in [AF 1]. Such a varifold is defined to be a Radon measure on the bundle over M whose fiber at each point p of M is the Grassmann manifold of k dimensional linear subspaces of the tangent space to M at p. If V is a varifold in M, let I I V I I be the Radon measure on M obtained from V by ignoring the fiber variables. Naturally injected in the space of k dimensional varifolds in M is the set of k dimensional rectifiable subsets of M, which includes the set of k dimensional submanifolds of M as well as more general k dimensional surfaces in M. A k dimensional varifold in M is said to be rectifiable (integral) if it can be strongly approximated by a positive real (integral) linear combination of varifolds corresponding to continuously differentiable k dimensional submanifolds of M. To any classical k dimensional geometric object in M there corresponds a k dimensional integral varifold in M. If N is a smooth Riemannian manifold and F: M-e N is smooth, then F induces in a natural way a strongly continuous mapping F# of the k dimen-

Variational inequalities with lower dimensional obstacles

We study a two-dimensional variational inequality where the interior constraining function, or obstacle, is defined only on a segment of the domain. A simple proof of the existence and smoothness of the solution to this type of problem is available via the results of a recent paper by Lewy and Stampacchia. Important special cases include the Dirichlet Integral and the area integrand.

A regularity theorem for the first variation of the area integrand

A regularity theorem for the first variation of the area integrand