I will present a proof with some substantial details of the
Multiplicity One Conjecture in Min-max theory, raised by Marques
and Neves. It says that in a closed manifold of dimension between 3
and 7 with a bumpy metric, the min-max minimal...
In this talk I want to discuss two related questions about the
variational structure of the Yang-Mills functional in dimension
four. The first is the question of 'gap' estimates; i.e.,
determining an energy threshold below which any solution must
be...
A decades-old application of the second variation formula proves
that if the scalar curvature of a closed 3--manifold is bounded
below by that of the product of the hyperbolic plane with the line,
then every 2--sided stable minimal surface has area...
Abstract: Uryson k-width of a metric space X measures how close
X is to being k-dimensional. Several years ago Larry Guth proved
that if M is a closed n-dimensional manifold, and the volume of
each ball of radius 1 in M does not exceed a certain...
Abstract: For surfaces immersed into a compact Riemannian
manifold, we consider the curvature functional given by the $L^2$
integral of the second fundamental form. We discuss an an area
bound in terms of that functional, with application to the...
Abstract: We present new-curvature one-cycle sweepout estimates
in Riemannian geometry, both on surfaces and in higher dimension.
More precisely, we derive upper bounds on the length of
one-parameter families of one-cycles sweeping out essential...
Abstract : It is a joint work with G. Courtois, S. Gallot and
A.Sambusetti. We prove a compactness theorem for metric spaces with
anupper bound on the entropy and other conditions that will be
discussed.Several finiteness results will be drawn. It...
Abstract: For a given finite subset S of a compact Riemannian
manifold (M; g) whose Schouten curvature tensor belongs to a given
cone, we establish a necessary and
sufficient condition for the existence and uniqueness of a
conformal metric on $M...
Abstract: It is a basic open question in geometric measure
theory to understand regularity of a stationary integral varifold.
Even a.e. regularity remains an open question. The central issue is
analyzing the varifold near a point Z with a tangent...
