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Register a new userA floating body approach to Feffermans hypersurface measure
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Authors
David E. Barrett
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The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Feffermans invariant hypersurface measure.
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We investigate the Plateau and isoperimetric problems associated to Feffermans measure for strongly pseudoconvex real hypersurfaces in $mathbb C^n$ (focusing on the case $n=2$), showing in particular that the isoperimetric problem shares features of both the euclidean isoperimetric problem and the corresponding problem in Blaschkes equiaffine geometry in which the key inequalities are reversed. The problems are invariant under constant-Jacobian biholomorphism, but we also introduce a non-trivial modified isoperimetric quantity invariant under general biholomorphism.
We will show that any open Riemann surface $M$ of finite genus is biholomorphic to an open set of a compact Riemann surface. Moreover, we will introduce a quotient space of forms in $M$ that determines if $M$ has finite genus and also the minimal genus where $M$ can be holomorphically embedded.
We study a new construction of bodies from a given convex body in $mathbb{R}^{n}$ which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to $p$-affine surface areas. We show that these bodies are related to Ulams long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.
Given a projective symplectic manifold $M$ and a non-singular hypersurface $X subset M$, the symplectic form of $M$ induces a foliation of rank 1 on $X$, called the characteristic foliation. We study the question when the characteristic foliation is algebraic, namely, all the leaves are algebraic curves. Our main result is that the characteristic foliation of $X$ is not algebraic if $X$ is of general type. For the proof, we first establish an etale version of Reeb stability theorem in foliation theory and then combine it with the positivity of the direct image sheaves associated to families of curves.
Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.
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