The Structure of Gaussian Minimal Bubbles
- PDF / 691,856 Bytes
- 42 Pages / 439.37 x 666.142 pts Page_size
- 28 Downloads / 179 Views
The Structure of Gaussian Minimal Bubbles Steven Heilman1 Received: 8 March 2020 / Accepted: 26 September 2020 © Mathematica Josephina, Inc. 2020
Abstract It is shown that m disjoint sets with fixed Gaussian volumes that partition Rn with minimum Gaussian surface area must be (m − 1)-dimensional. This follows from a second variation argument using infinitesimal translations. The special case m = 3 proves the Double Bubble problem for the Gaussian measure, with an extra technical assumption. That is, when m = 3, the three minimal sets are adjacent 120 degree sectors. The technical assumption is that the triple junction points of the minimizing sets have polynomial volume growth. Assuming again the technical assumption, we prove the m = 4 Triple Bubble Conjecture for the Gaussian measure. Our methods combine the Colding–Minicozzi theory of Gaussian minimal surfaces with some arguments used in the Hutchings–Morgan–Ritoré-Ros proof of the Euclidean Double Bubble Conjecture. Keywords Gaussian · Bubble · Minimal surface · Calculus of variations Mathematics Subject Classification 60E15 · 53A10 · 60G15 · 58E30
1 Introduction Classical isoperimetric theory asks for the minimum total Euclidean surface area of m disjoint volumes in Rn+1 . The case m = 1 results in the Euclidean ball. That is, a Euclidean ball has the smallest Euclidean surface area among all (measurable) sets of fixed Lebesgue measure. The case m = 2 is the Double Bubble Problem, solved in [1,2]. The case m ≥ 3 is still open, except for the special case m = 3, n +1 = 2 [3]. As
Supported by NSF Grant DMS 1829383.
B 1
Steven Heilman [email protected] Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA
123
S. Heilman
Hutchings writes on his website1 concerning the m = 3, n + 1 = 3 case, “The triple bubble problem in R3 currently seems hopeless without some brilliant new idea.” Recent results in theoretical computer science, such as sharp hardness for the MAX-m-CUT problem [4,5] motivate the above isoperimetric problem with Lebesgue measure replaced with the Gaussian measure [5]. Also, the “plurality is stablest” conjecture from social choice theory is closely related to such an isoperimetric problem. This problem [5] says that if votes are cast in an election between m candidates, if every candidate has an equal chance of winning, and if no one person has a large influence on the outcome of the election, then taking the plurality is the most noise-stable way to determine the winner of the election. That is, plurality is the voting method where the outcome is least likely to change due to independent, uniformly random changes to the votes. The latter conjecture is a generalization of the “majority is stablest conjecture” [6], which was proven using (a generalization of) the Gaussian m = 1 case of the isoperimetric problem posed above. In the Gaussian setting, for convenience, we include the complement of the m volumes as a set itself. That is, in the Gaussian setting, we ask for the minimum total Gaussian
Data Loading...
