Almgren

Bio: Almgren is an academic researcher. The author has contributed to research in topics: Almost everywhere. The author has an hindex of 1, co-authored 1 publications receiving 408 citations.

The motion of a surface by its mean curvature

Abstract: Kenneth Brakke studies in general dimensions a dynamic system of surfaces of no inertial mass driven by the force of surface tension and opposed by a frictional force proportional to velocity. He formulates his study in terms of varifold surfaces and uses the methods of geometric measure theory to develop a mathematical description of the motion of a surface by its mean curvature. This mathematical description encompasses, among other subtleties, those of changing geometries and instantaneous mass losses.Originally published in 1978.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

The inverse mean curvature flow and the Riemannian Penrose Inequality

Abstract: Let M be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface N in M is bounded by the ADM mass m according to the formula |N |≤ 16πm 2 . We develop a theory of weak solutions of the inverse mean curvature flow, and employ it to prove this inequality for each connected component of N using Geroch’s monotonicity formula for the ADM mass. Our method also proves positivity of Bartnik’s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry.

Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory

Abstract: The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.

The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces

Abstract: In this paper we provide a complete classification of the local structure of singularities in a wide class of two-dimensional surfaces in R3 collected under the adjective (M, i, a) minimal by Almgren [A3] (see I(8)). The results, Theorems II. 4, IV. 5, IV. 8, are that the singular set of an (M, i, a) minimal set consists of H6lder continuously differentiable curves along which three sheets of the surface meet (Holder continuously) at equal (120?) angles, together with isolated points at which four such curves meet bringing together six sheets of the surface (H6lder continuously) at equal anglesin fact, in a neighborhood of each singular point, the surface is H6lder continuously diffeomorphic to either the surface Y of Figure 1 or the surface T of Figure 2 (both of which are defined in I(11)). The results apply to (idealizations of) many actual surfaces which are governed by surface tension, such as soap films as in Figure 4 and compound soap bubbles as in Figure 3 (and therefore to aggregates of some kinds of biological and metallurgical cells) (Corollary IV. 9 (i), (ii)), and thus are a proof of a result deduced experimentally by Plateau over 100 years ago [P]. They also apply to surfaces which minimize integrals which equal the area integral times some Holder continuous function on R3. A necessary first step in classifying singularities is to determine all possible area minimizing cones (Proposition II. 3). (In 1864 Lamarle claimed to make such a determination but his analysis of the technically most difficult case Figure 12 (p. 503)-was wrong.) Also included is a proof that the surface T of Figure 2 is in fact area minimizing (Theorem IV. 6); it seems to require the full force of Theorem IV.5 and I have never seen it proved elsewhere. The methods of this paper are