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The volume of a unit vector field in 2 dimensions via calibrations

79   0   0.0 ( 0 )
 Added by Rui Albuquerque
 Publication date 2021
  fields
and research's language is English




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We use the theory of calibrations to write the equation of a minimal volume vector field on a given Riemann surface.

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59 - Pierre Dehornoy 2016
We construct a new invariant-the trunkenness-for volume-perserving vector fields on S^3 up to volume-preserving diffeomorphism. We prove that the trunkenness is independent from the helicity and that it is the limit of a knot invariant (called the trunk) computed on long pieces of orbits.
66 - Chang Li , Lei Ni , 2020
By studying a complex Monge-Amp`ere equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact Kahler manifold $N^n$ with $Ric_k< 0$ for some integer $k$ with $1<k<n$, and the ampleness of the canonical line bundle $K_N$.
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We give a new CR invariant treatment of the bigraded Rumin complex and related cohomology groups via differential forms. We also prove related Hodge decomposition theorems. Among many applications, we give a sharp upper bound on the dimension of the Kohn--Rossi groups $H^{0,q}(M^{2n+1})$, $1leq qleq n-1$, of a closed strictly pseudoconvex manifold with a contact form of nonnegative pseudohermitian Ricci curvature; we prove a sharp CR analogue of the Frolicher inequalities in terms of the second page of a natural spectral sequence; and we generalize the Lee class $mathcal{L}in H^1(M;mathscr{P})$ -- whose vanishing is necessary and sufficient for the existence of a pseudo-Einstein contact form -- to all nondegenerate orientable CR manifolds.
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New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harmonic maps from 3-dimensional Euclidean domains to the 2-sphere. Finally, the harmonic unit vector field tangent to a parallel family of hyperbolic geodesics is shown to be unstable, by constructing a class of compactly supported energy-decreasing variations. All examples considered have infinite total bending.
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