51 research outputs found
Canonical Sasakian Metrics
Let be a closed manifold of Sasaki type. A polarization of is defined
by a Reeb vector field, and for one such, we consider the set of all Sasakian
metrics compatible with it. On this space, we study the functional given by the
squared -norm of the scalar curvature. We prove that its critical points,
or canonical representatives of the polarization, are Sasakian metrics that are
transversally extremal. We define a Sasaki-Futaki invariant of the
polarization, and show that it obstructs the existence of constant scalar
curvature representatives. For a fixed CR structure of Sasaki type, we define
the Sasaki cone of structures compatible with this underlying CR structure, and
prove that the set of polarizations in it that admit a canonical representative
is open.Comment: 36 pages, minor corrections made, example adde
Conformal invariants of isometric embeddings of the smooth metrics on a surface
We view all smooth metrics on a closed surface through their
Nash isometric embeddings into a standard sphere of large, but fixed, dimension .
We define the Willmore functional over this space of
metrics on in terms of the extrinsic quantities of . Its infimum
over metrics in a conformal class is an invariant of the class varying
differentiably with it. If is oriented of genus , when , we
use the gap theorem of Simons to show that there is a unique conformal class of
metrics on , whose invariant is the value for the standard
totally geodesic embedding of , and we have that , with the lower bound achieved if, and only if, is conformally
equivalent to this standard geodesic embedding of , and , while when , the Lawson minimal surface
fixes the scale, and we show that
,
with the lower bound achieved by if, and only if, is conformally
equivalent to ,
and . For a
nonoriented , we prove a likewise estimate from below for
, and characterize conformally the surface that
realizes the optimal lower bound
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