Mathematics

If ?: M n ??R n+1 is a smooth immersed closed hypersurface, we consider the functional F m (?)= ??M 1+| ??m ν | 2 dμ , where ν is a local unit normal vector along ? , ??is the Levi-Civita connection of the Riemannian manifold (M,g) , with g the pull-back metric induced by the immersion and μ the associated volume measure. We prove that if m>?�n/2??then the unique globally defined smooth solution to the L 2 -gradient flow of F m , for every initial hypersurface, smoothly converges asymptotically to a critical point of F m , up to diffeomorphisms. The proof is based on the application of a Lojasiewicz-Simon gradient inequality for the functional F m .

In this note, we shall investigate the asymptotic curvature estimate on steady Ricci solitons.

We extend V. Arnold's theory of asymptotic linking for two volume preserving flows on a domain in R 3 and S 3 to volume preserving actions of R k and R ℓ on certain domains in R n and also to linking of a volume preserving action of R k with a closed oriented singular ℓ -dimensional submanifold in R n , where n=k+ℓ+1 . We also extend the Biot-Savart formula to higher dimensions.

We discuss asymptotically hyperbolic manifold with a noncompact boundary which is close to a horosphere in a certain sense. The model case is a horoball or the complement of a horoball in standard hyperbolic space. We show some geometric formulas.

We show that a Kähler-Ricci soliton on a Fano manifold can always be smoothly approximated by a sequence of relative anticanonically balanced metrics, also called quantized Kähler-Ricci solitons. The proof uses an equivariant version of Berezin-Toeplitz quantization to extend a strategy due to Donaldson, and can be seen as the quantization of a method due to Tian and Zhu, using quantized Futaki invariants as obstructions for quantized Kähler-Ricci solitons. As a by-product, we show that a Kähler-Einstein Fano manifold does not necessarily admit anticanonically balanced metrics in the usual sense when its automorphism group is not discrete.

Fedosov used flat sections of the Weyl bundle on a symplectic manifold to construct a star product ⋆ which gives rise to a deformation quantization. By extending Fedosov's method, we give an explicit, analytic construction of a sheaf of Bargmann-Fock modules over the Weyl bundle of a Kähler manifold X equipped with a compatible Fedosov abelian connection, and show that the sheaf of flat sections forms a module sheaf over the sheaf of deformation quantization algebras defined ( C ∞ X [[ℏ]],⋆) . This sheaf can be viewed as the ℏ -expansion of L ⊗k as k→∞ , where L is a prequantum line bundle on X and ℏ=1/k .

Using log canonical thresholds and basis divisors Fujita--Odaka introduced purely algebro-geometric invariants δ m whose limit in m is now known to characterize uniform K-stability on a Fano variety. As shown by Blum-Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these δ m -invariants characterizes uniform Ding stability. A basic question since Fujita-Odaka's work has been to find an analytic interpretation of these invariants. We show that each δ m is the coercivity threshold of a quantized Ding functional on the m -th Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for P n . Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kähler-Ricci solitons (and the more general g -solitons of Berman-Witt Nyström), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.

In this paper we consider the Balmuş-Montaldo-Oniciuc's conjecture in the case of hemispheres. We prove that a compact non-minimal biharmonic hypersurface in a hemisphere of S n+1 must be the small hypersphere S n (1/ 2 – √ ) , provided that n 2 − H 2 does not change sign.

Using the method of blow-up analysis, we obtain two sharp Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary, as well as the existence of the corresponding extremals. This generalizes early results of Chang-Yang [7] and the first named author [32], and complements Fontana's inequality of two dimensions [15]. The blow-up analysis in the current paper is far more elaborate than that of [32], and particularly clarifies several ambiguous points there. In precise, we prove the existence of isothermal coordinate systems near the boundary, the existence and uniform estimates of the Green function with the Neumann boundary condition. Also our analysis can be applied to the Kazdan-Warner problem and the Chern-Simons Higgs problem on compact Riemman surfaces with smooth boundaries.

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.