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SPHERES OF PRESCRIBED MEAN CURVATURE IN S3 MICHAEL T. ANDERSON
 

Summary: SPHERES OF PRESCRIBED MEAN CURVATURE IN S3
MICHAEL T. ANDERSON
Abstract. We prove a "semi-global" result on the existence of conformal embeddings of S2
into
S3
of prescribed mean curvature.
1. Introduction
In this note, we discuss the existence of conformal embeddings of a two-sphere S2 with prescribed
mean curvature into the round 3-sphere S3 of radius 1.
To state the main result, let C1 = Cm-1,
1 denote the space of Cm-1, functions H : S2 R
satisfying
0 < H < 1
where m 3, (0, 1). Define an equivalence relation on C1 by
(1.1) [H1] = [H2] H2 = H1 + ,
where = cixi is the restriction of a linear function to S2 R3, so that the gradients form
the space of conformal vector fields on S2. Let D1 = C1/ be the quotient space.
Theorem 1.1. For any pointwise Cm, conformal class [] of metrics on S2 and for any equivalence
class [H] D1, there exists a Cm+1, embedding of S2 into S3 with prescribed pointwise conformal
class [] and prescribed mean curvature class [H]. Thus, there exists a Cm+1, smooth embedding

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics