arXiv research

New Ricci flows found with Einstein orbifolds at infinity.

We study the long time behaviour of Ricci flow with bubbling-off on a possibly noncompact $3$-manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston's hyperbolisation theorem for $3$-manifolds with toral boundary that generalizes Perelman's proof …

In joint work with Chen and Weber, the author has elsewhere shown that CP2#2(-CP2) admits an Einstein metric. The present paper gives a new and rather different proof of this fact. Our results include new existence theorems for extremal Kahler metrics, and these allow one to prove the above existence statement by defor…

Study curvature measures of surface sequences in 4-manifolds converging to branched surfaces.

In this paper we will prove that for a compact, symplectic manifold $(M, ω)$ and for $ω$-compatible almost-complex structure J any properly perturbed J-holomorphic curve has a non-negative symplectic area. This non-negative property provides us with a new obstruction to the bubbling off phenomenon and thus allows us to…

Continuous process closes cusps in complex algebraic surfaces.

Let $(M,g)$ be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point $p\in M$ is called the mass endomorphism in $p$ associated to the metric $g$ due to an analogy to the mass in the Yamabe problem. We show that the mass endomorphism of a gen…

In this article, we introduce a mass-decreasing flow for asymptotically flat three-manifolds with nonnegative scalar curvature. This flow is defined by iterating a suitable Ricci flow with surgery and conformal rescalings and has a number of nice properties. In particular, wormholes pinch off and nontrivial spherical s…

In this paper we study ${\rm Spin}(7)$-instantons on asymptotically conical ${\rm Spin}(7)$-orbifolds (and manifolds) obtained by filling in certain squashed $3$-Sasakian $7$-manifolds. We construct a $1$-parameter family of explicit ${\rm Spin}(7)$-instantons. Taking the parameter to infinity, the family (a) bubbles o…

Kaehler-Einstein metrics on orbifolds derived from Einstein sequences.

In this paper (S_n) is a sequence of surfaces immersed in a 4-manifold which converges to a branched surface S_0. Up to sign, μ^T_p (resp. μ^N_p) will denote the amount of curvature of the tangent bundles TS_n (resp. the normal bundles NS_n) which concentrates around a singular point p of S_0 when n goes to infinity. B…

Study on half spheres solves Nirenberg problem with complex blow-up analysis.

We prove that a sequence of Fueter sections of a bundle of compact hyperkahler manifolds $\mathfrak X$ over a $3$-manifold $M$ with bounded energy converges (after passing to a subsequence) outside a $1$-dimensional closed rectifiable subset $S \subset M$. The non-compactness along $S$ has two sources: (1) Bubbling-off…

The main topic is the development of a Fredholm theory in a new class of spaces called M-polyfolds. In the subsequent Volume II the theory will be generalized to an even larger class of spaces called polyfolds, which can also incorporate local symmetries. The whole package provides a functional analytic framework to de…

Proves Weinstein's and Arnold's conjectures using contact instantons.

We construct a compactification of the moduli space of twisted holomorphic maps with varying complex structure and bounded energy. For a given compact symplectic manifold $X$ with a compatible complex structure and a Hamiltonian action of $S^1$ with moment map $μ:X\to\imag\RR$, the moduli space which we compactify cons…