The monotonicity formula for the area functional (and for other important functionals enjoying enough symmetry, such as the Dirichlet energy for maps, the Yang-Mills energy for connections, etc) is a basic tool used crucially in the proof of a number of fundamental facts, including the upper semicontinuity of the support for a sequence of stationary varifolds (with a lower density bound), the compactness of stationary rectifiable varifolds and the existence of tangent cones. The first two are particularly mportant in soft arguments by compactness and contradiction. Here we discuss the anisotropic version of the Michael-Simon inequality, which we can prove for (strictly) convex anisotropic integrands for two-dimensional varifolds in R^3, provided that the integrand is close to the area in the C^1 topology. This inequality, which we conjecture to hold even without the closeness constraint, is good enough to obtain lower area bounds when the density of the varifold is bounded below. This allows to recover upper semicontinuity of the support, compactness of rectifiable and integral varifolds, and it also allows to prove Allard's regularity theorem in this setting (using also recent work by De Rosa-Tione), where the monotonicity formula is probably false. The proof is deeply inspired by posthumous notes by Almgren, devoted to the same result. Although our arguments overlap with Almgren's, some parts are greatly simplified and rely on a nonlinear inequality bounding the L^1-norm of the determinant of a function, from the plane to 2x2 matrices, with the L^1-norms of the divergence of the rows, provided the matrix obeys (pointwise) some nonlinear constraints. This inequality can be seen as a version of the multilinear Kakeya inequality on the plane. This is joint work with Guido De Philippis (NYU).
Surviving without monotonicity: anisotropic Michael-Simon inequality
Pigati, Alessandro;
2021
Abstract
The monotonicity formula for the area functional (and for other important functionals enjoying enough symmetry, such as the Dirichlet energy for maps, the Yang-Mills energy for connections, etc) is a basic tool used crucially in the proof of a number of fundamental facts, including the upper semicontinuity of the support for a sequence of stationary varifolds (with a lower density bound), the compactness of stationary rectifiable varifolds and the existence of tangent cones. The first two are particularly mportant in soft arguments by compactness and contradiction. Here we discuss the anisotropic version of the Michael-Simon inequality, which we can prove for (strictly) convex anisotropic integrands for two-dimensional varifolds in R^3, provided that the integrand is close to the area in the C^1 topology. This inequality, which we conjecture to hold even without the closeness constraint, is good enough to obtain lower area bounds when the density of the varifold is bounded below. This allows to recover upper semicontinuity of the support, compactness of rectifiable and integral varifolds, and it also allows to prove Allard's regularity theorem in this setting (using also recent work by De Rosa-Tione), where the monotonicity formula is probably false. The proof is deeply inspired by posthumous notes by Almgren, devoted to the same result. Although our arguments overlap with Almgren's, some parts are greatly simplified and rely on a nonlinear inequality bounding the L^1-norm of the determinant of a function, from the plane to 2x2 matrices, with the L^1-norms of the divergence of the rows, provided the matrix obeys (pointwise) some nonlinear constraints. This inequality can be seen as a version of the multilinear Kakeya inequality on the plane. This is joint work with Guido De Philippis (NYU).
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