Communications in Analysis and Geometry

Volume 26 (2018)

Number 3

On the compactness theorem for embedded minimal surfaces in $3$-manifolds with locally bounded area and genus

Pages: 659 – 678



Brian White (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity, and we analyze what happens when one blows up the surfaces near a point where the convergence is not smooth.

2010 Mathematics Subject Classification

Primary 53A10. Secondary 49Q05.

The author’s research was supported by NSF grants DMS-1105330 and DMS-1404282.

Received 5 April 2015

Published 27 July 2018