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Mathematics, Differential Geometry, math.DG,Mathematics, Analysis of PDEs, math.AP,Mathematics, Metric Geometry, math.MG,
Abstract:
A second fundamental form is introduced for arbitrary closed subsets of
Euclidean space, extending the same notion introduced by J. Fu for sets of
positive reach. We extend well known integral-geometric formulas to this
general setting and we provide a structural result in terms of second
fundamental forms of submanifolds of class $2$ that is new even for sets of
positive reach. In the case of a large class of minimal submanifolds, which
include viscosity solutions of the minimal surface system and rectifiable
stationary varifolds of arbitrary codimension and higher multiplicities, we
prove the area formula for the generalized Gauss map in terms of the
discriminant of the second fundamental form and, adapting techniques from the
theory of viscosity solutions of elliptic equations to our geometric setting,
we conclude a natural second-order-differentiability property almost
everywhere. Moreover the trace of the second fundamental form is proved to be
zero for stationary integral varifolds.
