The Morse index theorem for regular Lagrangian systems

p-Yang-Mills 连接的莫尔斯指数稳定性

Morse index stability for p-Yang-Mills connections

本文研究了p-Yang-Mills连接在四维空间中Sacks-Uhlenbeck松弛下的临界点序列的莫尔斯指数稳定性，揭示了Yang-Mills场相较于谐调映射的更稳定行为。

Quantum corrections to the holomorphic structure of the mirror bundle along the caustic and the bifurcation locus

Given, in the Lagrangian torus fibration $R^4\to R^2$, a Lagrangian submanifold $L$, endowed with a trivial flat connection, the corresponding mirror object is constructed on the dual fibration by means of a family of Morse homologies associated to the generating function of $L$, and it is provided with a holomorphic structure. Morse homology, however, is not defined along the caustic $C$ of $L$ or along the bifurcation locus $B$, where the family does not satisfy the Morse-Smale condition. The holomorphic structure is extended to the subset $C\cup B$, except cusps, yielding the so called quan…

An Index Theorem for Non Periodic Solutions of Hamiltonian Systems

We consider a {\em Hamiltonian setup} $\sextuple$, where $(\mathcal M,ω)$ is a symplectic manifold, $\mathfrak L$ is a distribution of Lagrangian subspaces in $\mathcal M$, $\mathcal P$ a Lagrangian submanifold of $ \mathcal M$, $H$ is a smooth time dependent Hamiltonian function on $\mathcal M$ and $Γ:[a,b]\to\mathcal M$ is an integral curve of the Hamiltonian flow $\Hf$ starting at $\mathcal P$. We do not require any convexity property of the Hamiltonian function $H$. Under the assumption that $Γ(b)$ is not $\mathcal P$-focal it is introduced the Maslov index $\maslov(Γ)$ of $Γ$ given in ter…

Cuplength estimates in Morse cohomology

The main goal of this paper is to give a unified treatment to many known cuplength estimates. As the base case, we prove that for $C^0$-perturbations of a function which is Morse-Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we work with rather general assumptions the proof also applies in a variety of Floer settings. For example, this proves lower bounds for the number of fixed points of Hamiltonian diffeomorphisms, Hamiltonian chords for Lagrangian submanifolds, translated points of contactomorphisms…

A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furthermore that they must be word hyperbolic. We introduced this class of discrete subgroups in our earlier paper, in the context of symmetric spaces, where various equivalent geometric and dynamical characterizations of word hyperbolic Morse subgroups were established, including th…

Finite Morse index implies finite ends

We prove that finite Morse index solutions to the Allen-Cahn equation in $\R^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularity of clustering interfaces for the singularly perturbed Allen-Cahn equation in $\R^n$. Using an indirect blow-up technique, in the spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show that the {\bf obstruction} to the uniform second order regularity of cl…

Twisted Novikov homology and circle-valued Morse theory for knots and links

The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). Novikov homology provides lower bounds for MN(L). In the present paper we introduce the notion of twisted Novikov homology, which allows to obtain better lower bounds for MN(L) than the usual Novikov homology. Our twisted Novikov homology is a module over the Novikov ring Z((t)) bu…

Conformal symplectic geometry of cotangent bundles

We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $β$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $β$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.

Schubert calculus on the grassmannian of hermitian lagrangian spaces

The grassmannian of hermitian lagrangian spaces in $\mathbb{C}^n\oplus \mathbb{C}^n$ is a natural compactification of the space of hermitian $n\times n$ matrices. We describe a Schubert-like, Whitney regular stratification on this space which has a Morse theoretic origin. We prove that these strata define closed subanalytic currents à la R. Hardt, generating the integral homology of this space, we investigate their intersection theoretic properties, and we prove certain odd (in K-theoretic sense) Thom-Porteous type theorems.

Willmore浸入的Morse指标稳定性 I

Morse Index Stability of Willmore Immersions I

本文借助Da Lio–Gianocca–Rivière的上半连续性方法，证明在无分支点条件下Willmore浸入的Morse指标与零化度之和上半连续，并涵盖特定退化Riemann曲面情形。

Lagrangian correspondences and Donaldson's TQFT construction of the Seiberg-Witten invariants of 3-manifolds

Using Morse-Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3-manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed intersections of such Lagrangians, we supply the analytic details of Donaldson's TQFT construction of the Seiberg-Witten invariants of a closed 3-manifold.

Characterizing Regular Lagrangians by Lefschetz Fibration

We prove that the existence of regular Lagrangians can be given by existence of Weinstein Lefschetz fibrations with an hypothesis.

分支Willmore浸入的Morse指数稳定性

Morse Index Stability of Branched Willmore Immersions

本文研究了具有有限能量的Willmore浸入的Morse指数稳定性，通过分析四阶微分算子的性质和Gagliardo-Nirenberg-Rellich不等式，证明了其下半连续性。

Lagrangian平均曲率方程凸粘性解的正则性

Regularity for convex viscosity solutions of Lagrangian mean curvature equation

本文研究Lagrangian平均曲率方程凸粘性解的正则性，在相位二阶导数Hölder连续的条件下证明其解为正则解。

Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions

For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\geq 2$, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal $G$-bundle over $X$ supporting a connection with $L^p$-small curvature, when $p>d/2$, to the case of a connection with $L^{d/2}$-small curvature. We prove an optimal Lojasiewicz-Simon gradient inequality for abstract Morse-Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis in arXiv:1510.03817. We apply this result to prove the optimal Lojasiewicz-Si…

Symmetric Reduction of Regular Controlled Lagrangian System with Momentum Map

In this paper, following the ideas in Marsden et al.[18], we set up the regular reduction theory of a regular controlled Lagrangian (RCL) system with symmetry and momentum map, by using Legendre transformation and Euler-Lagrange vector field, and this reduction is an extension of symmetric reduction theory of a regular Lagrangian system under regular controlled Lagrangian equivalence conditions. Considering the completeness of reduction, in order to describe uniformly the RCL systems defined on a tangent bundle and on its regular reduced spaces, we first define a kind of RCL systems on a sympl…

On the Regularity of Hamiltonian Stationary Lagrangian manifolds

We prove a Morrey-type theorem for Hamiltonian stationary submanifolds of $\mathbb{C}^{n}$. Namely, if $L$ $\subset$ $\mathbb{C}^{n}$ is a $C^{1}$ Lagrangian submanifold with weakly harmonic Lagrangian phase $θ,$ then $L$ must be smooth. In the process we also discuss a local version of the equation, which is a nonlinear fourth order double divergence equation of the potential function whose gradient graph defines the Hamiltonian stationary submanifolds locally, and we establish full regularity and removability of singular sets of capacity zero for weak solutions with $C^{1,1}$ norm below a di…

奇异时间依赖拉格朗日系统的正则化

Regularization of singular time-dependent Lagrangian systems

本文基于共异弦嵌入定理，提出了一种新的正则化方法，结合Tulczyjew同构和几乎积结构，扩展了时间依赖奇异拉格朗日系统的正则化理论，并证明了一阶正则化的唯一性。

Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces

We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasiewicz proved by Simon (1983). We also prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the functional is Morse-Bott, improving on similar results due to Chill (2003, 2006), Haraux and Jendoubi (2007), and Simon (1996). In our article ar…