Narrow Results

Let T be a positive closed current of bidegree (1,1) on a compact complex surface. We show that for all ε > 0, one can find a finite composition of blow-ups π such that π*T decomposes as the sum of a divisorial part and a positive closed current whose Lelong numbers are all less than ε.

In the present paper the small quantum cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from ℙ³ or the quadric Q³ is explicitely computed. Because of systematic usage of the associativity property of quantum product only a very small and enumerative subset of Gromov–Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of quantum cohomology is proven by checking the computed quantum cohomology rings and by showing that a smooth Fano threefold X with b₃(X) = 0 admits a complete exceptional set of the appropriate length.

Let f be a positive smooth function on a closed Riemann surface (M, g). The f-energy of a map u from M to a Riemannian manifold (N, h) is defined as

In this paper we first give an integral condition under which the mean curvature flow can be extended in arbitrary codimension. Then we investigate some properties of Type I singularity.

In this paper, we investigate the initial boundary value problem of the nonlinear fourth-order dispersive-dissipative wave equation. By using the concavity method, we establish a blow-up result for certain solutions with arbitrary positive initial energy.

The blow-up analysis for a sequence of exponentially harmonic maps from a closed surface is studied to reestablish an existence result of harmonic maps from a closed surface into a closed manifold whose 2-dimensional homotopy class vanishes.

In this paper, we study the asymptotic behavior of Palais-Smale sequences associated with the prescribed mean curvature equation on a compact manifold with boundary. We prove that every such sequence converges to a solution of the associated equation plus finitely many “bubbles” obtained by rescaling fundamental solutions of the corresponding Euclidean boundary value problem.

This paper presents a new non-local expanding flow for convex closed curves in the Euclidean plane which increases both the perimeter of the evolving curves and the enclosed area. But the flow expands the evolving curves to a finite circle smoothly if they do not develop singularity during the evolving process. In addition, it is shown that an additional assumption about the initial curve will ensure that the flow exists on the time interval $[0,\infty )$. Meanwhile, a numerical experiment reveals that this flow may blow up for some initial convex curves.

In this paper, we study the dynamical behavior of solutions of nonlinear Schrödinger equations with quadratic interaction and $L^2$-critical growth. We give sharp conditions under which the existence of global and blow-up solutions are deduced. We also show the existence, stability, and blow-up behavior of normalized solutions of this system.