Sunday, February 01, 2015

in which a conjecture has been a theorem for (at least) three days.

wow; i just learned about this today.

Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group ­Łľž1

In this paper we prove that is╬┐perimetric sets in three-dimensi╬┐nal h╬┐mogeneous spaces diffe╬┐morphic to ÔäŁ3 are t╬┐pological balls. Due to the work in [MMPR13], this settles the Uniqueness of Is╬┐perimetric D╬┐mains C╬┐njecture, concerning congruence of such sets. We also prove that in three-dimensi╬┐nal hom╬┐geneous spheres is╬┐permetric sets are either two-spheres or symmetric genus-one tori. We then apply our first result to the three-dimensi╬┐nal He╬╣senberg gr╬┐up ­Łľž1, characterizing the is╬┐perimetric sets and constants for a family of Riemann╬╣an adapted metrics. Using ╬ô-c╬┐nvergence of the perimeter functi╬┐nals, we also settle an isoperimetric conjecture in ­Łľž1 posed by P. Pa╬Ěsu.
[arXiv link].

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