We study the symplectic structure of the holomorphic coadjoint orbits, generalizing a theorem of McDuff on the symplectic structure of Hermitian symmetric spaces of noncompact type.

Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface is at least that of the self-shrinking two-sphere. We prove this conjecture for all closed embedded $2$-spheres. Assuming a conjectural Morse index bound (announced recently by Marques-Neves), we can improve the result to apply to all closed embedded surfaces that are not tori. Our results can be thought of as the parabolic analog to the Willmore conjecture and our argument is analogous in many ways to that of Marques-Neves on the Willmore problem. The main tool is the min-max theory applied to the Gaussian area functional in $\mathbb{R}^3$ which we also establish. To any closed surface in $\mathbb{R}^3$ we associate a four parameter canonical family of surfaces and run a min-max procedure. The key step is ruling out the min-max sequence approaching a self-shrinking plane, and we accomplish this with a degree argument. To establish the min-max theory for $\mathbb{R}^3$ with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of non-stationary varifolds (with respect to the Gaussian metric of $\mathbb{R}^3$) in a continuous manner.

We study Dirac structures associated with Manin pairs (d, g) and give a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach [3] in terms of quasi-Poisson structures. We explain how these two distinct frameworks are related to each other, proving that they lead to isomorphic categories of Hamiltonian spaces. We stress the connection between the viewpoint of Dirac geometry and equivariant differential forms. The paper discusses various examples, including q-Hamiltonian spaces and Poisson-Lie group actions, explaining how presymplectic groupoids are related to the notion of “double” in each context.

We study the Kahler-Ricci flow on a class of projective bundles over a compact Kahler-Einstein manifold. Assuming the initial Kahler metric admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple (Σ,L,[ω0]), under which the P1-fiber collapses along the Kahler-Ricci flow and the projective bundle converges to Σ in the Gromov-Hausdorff sense. Furthermore, the Kahler-Ricci flow must have Type I singularity and is of (Cn ×P1)-type. This generalizes and extends part of Song-Weinkove’s work on Hirzebruch surfaces.

For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,∂M) where 3≤(n+1)≤7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates to prove that for almost every (in the C-infinity Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If ∂M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.