A generalized maximum principle for a smooth vector field and its applications to weak star-shaped hypersurfaces
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To what extent is the classical maximum principle for scalar-valued functions valid for a vector field on a compact Riemannian manifold? In response, we derive a maximum principle for a smooth vector field with its divergence term on a closed, oriented Riemannian manifold by the conformal geometrical technique that do not widely known. We then use it to obtain some applications to star-shaped hypersurfaces in the Euclidean space and extend the previous spherical result of Hsiung \cite{Hsi54} to weak star-shaped hypersurfaces. More precisely, we show that an orientable, connected, compact, weak star-shaped hypersurface in the Euclidean space with constant higher order mean curvature is a sphere.
Abstract
To what extent is the classical maximum principle for scalar-valued functions valid for a vector field on a compact Riemannian manifold? In response, we derive a maximum principle for a smooth vector field with its divergence term on a closed, oriented Riemannian manifold by the conformal geometrical technique that do not widely known. We then use it to obtain some applications to star-shaped hypersurfaces in the Euclidean space and extend the previous spherical result of Hsiung \cite{Hsi54} to weak star-shaped hypersurfaces. More precisely, we show that an orientable, connected, compact, weak star-shaped hypersurface in the Euclidean space with constant higher order mean curvature is a sphere.