290 research outputs found
High-jet relations of the heat kernel embedding map and applications
For any compact Riemannian manifold and its heat kernel embedding map
from M into constructed in [BBG], we study the higher derivatives
of with respect to an orthonormal basis at on . As the heat flow
time goes to 0, it turns out the limiting angles between these derivative
vectors are universal constants independent on , and the choice of
orthonormal basis. Geometric applications to the mean curvature and the
Riemannian curvature are given. Some algebraic structures of the infinite jet
space of are explored.Comment: 28 pages, related references on random functions are adde
Band width estimates via the Dirac operator
Let be a closed connected spin manifold such that its spinor Dirac
operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian
metric on with scalar curvature bounded below by , the distance between the boundary components of is at most
, where with being a universal constant. This verifies a conjecture of Gromov
for such manifolds. In particular, our result applies to all high-dimensional
closed simply connected manifolds which do not admit a metric of positive
scalar curvature. We also establish a quadratic decay estimate for the scalar
curvature of complete metrics on manifolds, such as ,
which contain as a codimension two submanifold in a suitable way.
Furthermore, we introduce the "-width" of a closed manifold and
deduce that infinite -width is an obstruction to positive scalar
curvature.Comment: 24 pages, 2 figures; v2: minor additions and improvements; v3: minor
corrections and slightly improved estimates. To appear in J. Differential
Geo
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