Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation

Abstract: We establish new lower bounds for the number of nodal bound states for the semiclassical nonlinear Schrodinger equation $$-\varepsilon^2 \Delta u+ a(x)u=|u|^{p-2}u$$ with bounded and uniformly continuous potential a. The solutions we obtain have precisely two nodal domains, and their positive and negative parts concentrate near the set of minimum points of a. Our approach is independent of penalization techniques and yields, in some cases, the existence of infinitely many nodal solutions for fixed $$\varepsilon$$ . Via a dynamical systems approach, we exhibit positively invariant sets of sign changing functions for the negative gradient flow of the associated energy functional. We analyze these sets on the cohomology level with the help of Dold’s fixed point transfer. In particular, we estimate their cuplength in terms of the cuplength of equivariant configuration spaces of subsets of $$\mathbb{R}^N$$ . We also provide new estimates of the cuplength of configuration spaces.

On realizing measured foliations via quadratic differentials of harmonic maps toR-trees

Abstract: We give a brief, elementary and analytic proof of the theorem of Hubbard and Masur [HM] (see also [K], [G]) that every class of measured foliations on a compact Riemann surfaceR of genusg can be uniquely represented by the vertical measured foliation of a holomorphic quadratic differential onR. The theorem of Thurston [Th] that the space of classes of projective measured foliations is a 6g—7 dimensional sphere follows immediately by Riemann-Roch. Our argument involves relating each representative of a class of measured foliations to an equivariant map from \( \tilde R \) to anR-tree, and then finding an energy minimizing such map by the direct method in the calculus of variations. The normalized Hopf differential of this harmonic map is then the desired differential.

Some 6-dimensional Hamiltonian S1-manifolds

Abstract: In an earlier paper we explained how to convert the problem of sym- plectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into CP 2 by using a new way to desingu- larize orbifold blow ups Z of the weighted projective space CP 2 1,m,n. We now use a related method to construct symplectomorphisms of these spaces Z. This allows us to construct some well known Fano 3-folds (including the Mukai-Umemura 3-fold) in purely symplectic terms using a classification by Tolman of a particular class of Hamiltonian S 1 -manifolds. We also show that (modulo scaling) these manifolds are uniquely determined by their fixed point data up to equivariant symplectomorphism. As part of this argument we show that the symplectomorphism group of a certain weighted blow up of a weighted projective plane is connected.

Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces

Abstract: We prove the conjectures of Graham-Kumar (GrKu08) and Griffeth- Ram (GrRa04) concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are im- mediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term—the top one—with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear.