Higher codimension isoperimetric problems

We consider a variational problem for submanifolds Q $\subset$ M with nonempty boundary $\partial$Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q \textless{} dim M . We then construct small nearly-spherical solutions of this higher codimension CMC prob-lem; these concentrate near the critical points of a certain curvature function

Refined asymptotics for constant scalar curvature metrics with isolated singularities

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, \Delta u + n(n-2)/4 u^{(n+2)(n-2) = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, by Caffarelli, Gidas and Spruck, we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohozaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions.Comment: To appear, Inventiones Mathematica

The Grizzly, February 20, 1987

Abortion Defended in Forum • College Ghosts Still a Mystery? • Letters: Campus Life Committee Defends Themselves; Art is not Such a Private Matter • Richter Announces UC Art Master Plan: Aggressive Couple to be Moved • Notes: Career Options Workshop; Non-credit Photo Class; Seminar of Tact Offered • Senior Dinner Season • Musser\u27s Second Year Plans Begin • Cub and Key Club Society Opens Doors • Bears on a Roll • Trio of Wins Key Bears\u27 Hopes to Battle Widener Saturday • Relay Drowns F&M • Family Affair: Karkoska and DeSimone • Last MAC Diving Championships • Ursinus to Host All-Star Track and Field Clinic • Athlete of the Week: Tricia Curryhttps://digitalcommons.ursinus.edu/grizzlynews/1182/thumbnail.jp