Smooth geometric evolutions of hypersurfaces

Smooth Geometric Evolutions of Hypersurfaces

We consider the gradient flow associated to the following functionals Fm(φ) = ∫ M 1 + |∇ν| dμ . The functionals are defined on hypersurfaces immersed in R via a map φ : M → R, where M is a smooth closed and connected n–dimensional manifold without boundary. Here μ and ∇ are respectively the canonical measure and the Levi–Civita connection on the Riemannian manifold (M, g), where the metric g is...

Smooth Divisors of Projective Hypersurfaces

Smooth Divisors of Projective Hypersurfaces

Star points on smooth hypersurfaces

— A point P on a smooth hypersurface X of degree d in PN is called a star point if and only if the intersection of X with the embedded tangent space TP (X) is a cone with vertex P . This notion is a generalization of total inflection points on plane curves and Eckardt points on smooth cubic surfaces in P3. We generalize results on the configuration space of total inflection points on plane curv...

Aleksandrov Reeection and Geometric Evolution of Hypersurfaces

Consider a compact embedded hypersurface ? t in R n+1 which moves with speed determined at each point by a function F (1 ; : : : ; n ; t) of its principal curvatures, for 0 t < T: We assume the problem is degenerate parabolic, that is, that F (; t) is nondecreasing in each of the principal curvatures 1 ; : : : ; n : We shall show that for t > 0 the hypersurface ? t sat-isses local a priori Lips...