13 research outputs found
Cohomogeneity-One Lagrangian Mean Curvature Flow
We study mean curvature flow of Lagrangians in that are
cohomogeneity-one under a compact Lie group acting
linearly on . Each such Lagrangian necessarily lies in a level
set of the standard moment map , and mean curvature flow preserves this containment. We
classify all cohomogeneity-one self-similarly shrinking, expanding and
translating solutions to the flow, as well as cohomogeneity-one smooth special
Lagrangians lying in . Restricting to the case of
almost-calibrated flows in the zero level set , we classify
finite-time singularities, explicitly describing the Type I and Type II blowup
models. Finally, given any cohomogeneity-one special Lagrangian submanifold in
, we show it occurs as the Type II blowup model of a Lagrangian
MCF singularity, thereby providing infinitely many previously unobserved
singularity models. Throughout, we give explicit examples of suitable group
actions, including a complete list in the case of simple.Comment: 53 pages, 3 figure
Calibrated Geometry in Hyperkahler Cones, 3-Sasakian Manifolds, and Twistor Spaces
We systematically study calibrated geometry in hyperk\"ahler cones
, their 3-Sasakian links , and the corresponding twistor
spaces , emphasizing the relationships between submanifold geometries
in various spaces. Our analysis emphasizes the role played by a canonical
-structure on the twistor space . We
observe that is an -family of
semi-calibrations, and make a detailed study of their associated calibrated
geometries.
As an application, we obtain new characterizations of complex Lagrangian and
complex isotropic cones in hyperk\"{a}hler cones, generalizing a result of
Ejiri and Tsukada. We also generalize a theorem of Storm on submanifolds of
twistor spaces that are Lagrangian with respect to both the K\"{a}hler-Einstein
and nearly-K\"{a}hler structures.Comment: 55 pages, 1 figure. Version 2: Added reference [3] to a paper of
Alexandrov that first observed this Sp(n)U(1) structure on the twistor space
A variational characterization of calibrated submanifolds
Let be a fixed compact oriented embedded submanifold of a manifold
. Consider the volume as a functional of the ambient metric on
, where . We show that is a
critical point of with respect to a special class of variations
of , obtained by varying a calibration on in
a particular way, if and only if is calibrated by . We do not assume
that the calibration is closed. We prove this for almost complex, associative,
coassociative, and Cayley calibrations, generalizing earlier work of Arezzo-Sun
in the almost K\"ahler case. The Cayley case turns out to be particularly
interesting, as it behaves quite differently from the others. We also apply
these results to obtain a variational characterization of Smith maps.Comment: 35 pages, comments welcom