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Summary: A SECOND ORDER SMOOTH VARIATIONAL PRINCIPLE ON
RIEMANNIAN MANIFOLDS
DANIEL AZAGRA AND ROBB FRY
Abstract. We establish a second order smooth variational principle valid
for functions defined on (possibly infinite-dimensional) Riemannian manifolds
which are uniformly locally convex and have a strictly positive injectivity ra-
dius and bounded sectional curvature.
1. Introduction and main result
It is well known that a continuous function defined on an infinite-dimensional
manifold (or on a Banach space) does not generally attain a minimum in situations
in which there would typically exist minimizers if the function were defined on a
finite-dimensional manifold (for instance when the infimum of the function in the
interior of a ball is strictly smaller than the infimum of the function on the boundary
of the ball). In fact, as shown in [1], the smooth functions with no critical points
are dense in the space of continuous functions on every Hilbert manifold (this result
may be viewed as a strong approximate version for infinite dimensional manifolds
of the Morse-Sard theorem). So, when we are given a smooth function on an
infinite-dimensional Riemannian manifold we should not expect to be able to find
any critical point, whatever the overall shape of this function is, as there might be
none.
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