Chaofeng Zhu
In this paper, we prove a Morse index theorem for the index form of even order linear Hamiltonian systems on the closed interval with reasonable self-adjoint boundary conditions. The highest order term is assumed to be nondegenerate.
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In this paper, we prove a Morse index theorem for the index form of regular Lagrangian system with selfadjoint boundary condition.
Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse index by its relative form, given by the spectral flow of an associated family of index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context,…
We give a new analytical proof of the Morse index theorem for geodesics in Riemannian manifolds.
We prove an extension of the Index Theorem for Morse-Sturm systems of the form $-V''+RV=0$, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M,g), we consider a geodesic $γ$ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, …
Morse Index bound of simple closed geodesics on 2-spheres and strong Morse Inequalities
本文利用Morse理论刻画黎曼二维球面上的简单闭测地线，证明在通用度量下存在Morse指标为1、2、3的简单闭测地线，并对可定向黎曼曲面上的简单闭曲线空间建立了长度泛函的强Morse不等式。
In this paper, we present a sufficient condition for finite Morse index of complete properly self-shrinkers. We prove that a complete properly embedded self-shrinker in $\mathbb{R}^{n+1}$ with finite asymptotically conical ends or asymptotically cylindrical ends must have finite Morse index. Moreover, as an application of this result, we show that a complete properly embedded self-shrinker in $\mathbb{R}^3$ with finite genus has finite Morse index.
We show that in a closed 3-manifold with a generic metric of positive Ricci curvature, there are minimal surfaces of arbitrary large Morse index, which partially confirms a conjecture by F. Marques and A. Neves. We prove this by analyzing the lamination structure of the limit of minimal surfaces with bounded Morse index.
The min-max construction of minimal spheres using harmonic replacement is introduced by Colding and Minicozzi and generalized by Zhou to conformal harmonic torus. We prove that the Morse index of the min-max conformal harmonic torus is bounded by one.
Given a closed manifold of dimension at least three, with non trivial homotopy group π_3(M) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bound one, with sum of their energies realizes a geometric invariant width.
We show that the Morse index of a closed minimal hypersurface in a four-dimensional Riemannian manifold cannot be bound in terms of the volume and the topological invariants of the hypersurface itself by presenting a method for constructing Riemannian metrics on S^4 that admit embedded minimal hyperspheres of uniformly bounded volume and arbitrarily large Morse index. The phenomena we exhibit are in striking contrast with the three-dimensional compactness results by Choi-Schoen.
We show that compact Riemannian three-manifolds with negative sectional curvature possess closed minimal surfaces of arbitrarily high Morse index.
We compute lower bounds for the Morse index and nullity of constant mean curvature tori of revolution in the three-dimensional unit sphere. In particular, all such tori have index at least five, with index growing at least linearly with respect to the number of the surfaces' bulges, and the index of such tori can be arbitrarily large.
We find various lower and upper bounds for the index of Wente tori that contain a continuous family of planar principal curves. We then prove a result that gives an algorithm for computing the index sharply.
In this work we give a new lower bound on the Morse index for constant mean curvature tori of revolution immersed in the three-sphere $\mathbb{S}^3$, by computing some explicit negative eigenvalues for the corresponding Jacobi operator.
On the Morse index of nonorientable minimal surfaces in $\mathbb{R}^3$
本文计算了三维欧氏空间中若干不可定向极小曲面的莫尔斯指标，证明Meeks莫比乌斯带指标为2、López克莱因瓶为3，并给出指标分类与下界结果。
We prove that any smooth harmonic map from $S^3$ into $S^2$ of Morse index less or equal than $4$ has to be an harmonic morphism, that is the successive composition of an isometry of $S^3$, the Hopf fibration and an holomorphic map from ${\mathbb C}P^1$ into itself.
We show that the rotationally symmetric free boundary minimal catenoid in the unit ball in $\Bbb{R}^3$ has Morse index equal to $4$.
Main theorem of this paper states that Floer cohomology groups in a Hilbert space are isomorphic to the cohomological Conley Index. It is also shown that calculating cohomological Conley Index does not require finite dimensional approximations of the vector field. Further directions are discussed.
Rigidity of Minimal Surfaces with Y-singularities and low Morse Index in $\mathbb{R}^3$
本文研究带 Y 奇点且低 Morse 指数的极小曲面，通过刚性结果给出指数为一情形的部分唯一性分类，突出 Y-catenoid 的特殊地位。
Morse index stability for Yang-Mills connections
本文证明了在弱收敛下4维Yang-Mills联络的Morse指数与零度之和的渐近上下界稳定性，界值由弱极限和气泡的Morse指数与零度构成。
The Morse index of constant curvature 2-spheres
计算了标准N-球面中所有具有常Gauss曲率的浸入极小2-球面的Morse指标和零化度，并给出了以Boruvka球面为链的R^7中结合锥的稳定性指标界。
We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using K-theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for families of selfadjoint elliptic operators.
Singularities of mean curvature flow with bounded mean curvature and Morse index
研究具有有界平均曲率和莫尔斯指数的平均曲率流奇点多重性，针对3≤n≤6时\(\mathbb{R}^{n + 1}\)中闭光滑嵌入流，证明首个奇异时间平均曲率或莫尔斯指数会爆炸。
We give three lower bounds for the Morse index of a constant mean curvature torus in Euclidean 3-space in terms of its spectral genus g. The first two lower bounds grow linearly in g and are stronger for smaller values of g, while the third grows quadratically in g but is weaker for smaller values of g.
We develop a bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids. We also provide bounds on the area of separating CMC surfaces of bounded (Morse) index and use this, together with the previous results, to bound their genus.
Revisiting Arnold's topological proof of the Morse index theorem
本文遵循并扩展 Arnold 的论文，利用 Maslov 指数阐述了 Riemannian 情形下的 Morse 指数定理，强调了证明中的 symplectic 论证并力求自洽。
In this paper we are concerned with harmonic maps and minimal immersions defined on compact Riemannian manifolds and with values in homogenous strongly harmonic manifolds. We show some results on the Morse index by varying these maps along suitable conformal vector fields. We obtain also that they are global maxima on some subspaces of the eigenspaces corresponding to the nonvanishing eigenvalues of the Laplacian operator on the target manifolds.
On the Morse Index with Constraints I: An Abstract Formulation
本文提出在希尔伯特空间中研究带约束的莫尔斯指数定理，并应用于闭合CMC超曲面、毛细面及类型II分区点的优化问题分析。
This is essentially a note on Section 7 of Perelman's first paper on Ricci flow. We list some basic properties of the index form for Perelman's $ \mathcal{L} $-length, which are analogous to the ones in Riemannian case (with fixed metric), and observe that Morse's index theorem for Perelman's $\mathcal{L}$-length holds. As a corollary we get the finiteness of the number of the $\mathcal{L}$-conjugate points along a finite $\mathcal{L}$-geodesic.
We show that the Morse index of every 2k-ended solution of the Allen-Cahn equation in R^2 is >= k-1. This bound is expected to be sharp.
We develop a general method to compute the Morse index of branched Willmore spheres and show that the Morse index is equal to the index of certain matrix whose dimension is equal to the number of ends of the dual minimal surface. As a corollary, we find that for all immersed Willmore spheres $\vecΦ:S^2\rightarrow \mathbb{R}^3$ such that $W(\vecΦ)=4πn$, we have $\mathrm{Ind}_{W}(\vecΦ)\leq n-1$
The $2^{nd}$ variation formula of the Seiberg-Witten functional is obtained in order to estimate the Morse index of redutible solutions $(A,0)$. It is shown that their Morse index is given by the dimension of the largest negative eigenspace of the operator $\triangle_{A} +\frac{k_{g}}{4}$, hence it is finite.
Morse index of min-max stationary integral varifolds
研究极小极大驻定整变分集的莫尔斯指数上界，针对闭合黎曼流形的d维p-宽度问题。
Minimal hypersurfaces of Morse index one
该研究证明了$\boldsymbol{R}^{n+1}$中满足有限全曲率、Morse指数为1等条件的极小超曲面的分类结果，确定其为高维悬链面。
The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the min-max minimal hypersurface. We advance the theory further and prove the first general Morse index bounds for minimal hypersurfaces produced by it. We also settle the multiplicity problem for the classical case of one-parameter sweepouts.
We give in this paper bounds for the Morse indices of a large class of simple geodesics on a surface with a generic metric. To our knowledge these bounds are the first that use only the generic hypothesis on the metric.
On any surface we give an example of a metric that contains simple closed geodesics with arbitrary high Morse index. Similarly, on any 3-manifold we give an example of a metric that contains embedded minimal tori with arbitrary high Morse index. Previously no such examples were known. We also discuss whether or not such bounds should hold for a generic metric and why bumpy does not seem to be the right generic notion. Finally, we mention briefly what such bounds might be used for.
The Morse index theorem in the case of two variable endpoints in conic Finsler manifolds
本文在基本张量沿测地线速度曲线正定的条件下，证明了具 $C^6$ 锥伪 Finsler 度量的 $C^7$ 流形中连接两子流形测地线的 Morse 指数定理。
We show that the sum of the Morse indices of the Willmore spheres realising the width of Willmore type sweep-outs is bounded by the number of the parameters of the min-max. As an application, we deduce that among the true Willmore spheres realising the min-max sphere eversion, at most one of them one has index 1, while the others must be stable.
Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and co-index. For any critical point x choose an integer a(x) arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the following property: for any pair of critical points x,y, the unstable manifold of x and the stable manifold of y have a transverse intersection of dimension a(x)-a(y).
Instability and Morse Index of Exponentially Subelliptic Harmonic Maps
研究指数次椭圆调和映射的Morse指标，通过外在和内在两种测试空间建立有限性并给出估计，适用于球面、球面乘积、凸超曲面及Heisenberg幂零流形。
Morse index and nullity of rotationally symmetric $Y$-singular minimal surfaces
本文在Y-奇异性构成单个圆周的假设下，通过固定边界问题和Dirichlet-to-Neumann映射计算旋转对称极小Y-奇异曲面的Morse指标，并证明Y-悬链面的指标为1。
The Morse index of quartic minimal hypersurfaces
针对$g=4$或$6$的$S^n$齐性极小超曲面Morse指标与零化度的未解问题，论文计算了两个$g=4$的此类超曲面的对应量，还发现其Laplace谱含不可根式表示的无理特征值。
Estimating the Morse index of free boundary minimal hypersurfaces through covering arguments
本文将Song的方法推广到自由边界情形，证明在3至7维带边紧致黎曼流形中，自由边界极小超曲面的Morse指标随Betti数之和线性增长，且增长常数依赖于其面积。
Given a Lorentzian manifold $(M,g)$, a geodesic $γ$ in $M$ and a timelike Jacobi field $\mathcal Y$ along $γ$, we introduce a special class of instants along $γ$ that we call $\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal Y$-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jaco…
Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of Fraser-Li, we conclude that the area index of a free-boundary minimal surface in a convex domain of Euclidean three-space, is bounded from above by a linear function of its genus and its number of boundary components. We also prove index bounds for submanifolds of higher dimensi…
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。
A smooth function f in a neighbourhood of the unit sphere $S^{n - 1}$ is said to admit index $λ$ if it can be extended to a function F in the unit ball $B^n$ such that F has a unique critical point p and the Morse index of p is equal to $λ$. It is easy to see that a function f cannot admit two indices of different parity. We prove that for any two indices that differ by two there exists a function f that admits both of them.
Morse index of free boundary disk for pseudoconvex domain
本文研究伪凸域或非负双截曲率 Kähler 流形中非全纯圆盘的 $\overline{\partial}$-能量 Morse 指标，证明能量极小化圆盘必为全纯，且非全纯临界圆盘的 Morse 指标至少为 $n-1$，并推广到 $k$-伪凸域。
We prove the upper-semi-continuity of the Morse index plus nullity of critical points to general conformally invariant Lagrangians in dimension 2 under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of an arbitrary sequence of weakly converging critical points to a general conformally invariant Lagrangians of maps from an arbitrary closed surface into an arbitrary closed smooth manifold passes to the limit in the following sense : it is asymptotically bounded from above by the sum of the Morse indices plus the nullity of the weak limit and the bubble…
In the previous work, the first author established an algorithm to compute the Morse index and the nullity of an $n$-periodic minimal surface in $\mathbb{R}^n$. In fact, the Morse index can be translated into the number of negative eigenvalues of a real symmetric matrix and the nullity can be translated into the number of zero-eigenvalue of a Hermitian matrix. The two key matrices consist of periods of the abelian differentials of the second kind on a minimal surface, and the signature of the Hermitian matrix gives a new invariant of a minimal surface. On the other hand, H family, rPD family, …
In this article we study the second variation of the energy functional associated to the Allen-Cahn equation on closed manifolds. Extending well known analogies between the gradient theory of phase transitions and the theory of minimal hypersurfaces, we prove the upper semicontinuity of the eigenvalues of the stability operator and consequently obtain upper bounds for the Morse index of limit interfaces which arise from solutions with bounded energy and index without assuming any multiplicity or orientability condition on these hypersurfaces. This extends some recent results of N. Le and F. Hi…
By extending and generalising previous work by Ros and Savo, we describe a method to show that the Morse index of every closed minimal hypersurface on certain positively curved ambient manifolds is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a relation between the index and the topology of minimal hypersurfaces holds, for example, on all compact rank one symmetric spaces, on products of the circle with spheres of arbitrary dimension and on suitably pinched submanifolds of the Euclidean spaces. These results confirm a ge…
On the Gauss Map of Anisotropic Minimal Surfaces and applications to the Morse Index estimates
本文通过证明各向异性极小曲面高斯映射临界集的离散性，将其视为到单位球面的分支覆盖，并利用共形几何技术给出莫尔斯指标的上、下界估计。
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…
The Morse Index, Nullity and Jacobi Fields of Constant-Curvature Minimal 2-Spheres in Complex Projective Spaces
该研究确定了高斯曲率为常数的极小浸入S²→ℂPᴺ的Morse指数与法零化度公式，还得到其全测地延拓至ℂPᴺ的对应公式，明确了法Jacobi核的相关性质。
We obtain an upper bound for the Morse index of Willmore spheres $Σ\subset S^3$ coming from an immersion of $S^2$. The quantization of Willmore energy shows that there exists an integer $m$ such that $\mathscr{W}(Σ)=4πm$. Then we show that $\mathrm{Ind}_{\mathscr{W}}(Σ)\leq m$. The proof relies on an explicit computation relating the second derivative of $\mathscr{W}$ for $Σ$ with the Jacobi operator of the minimal surface in $\mathbb{R}^3$ it is the image of by stereographic projection thanks of the fundamental classification of Robert Bryant.
The Morse index theorem for mechanical systems with reflections
该研究证明了可在超曲面处反射的路径上作用泛函的Morse指标定理，涵盖流形内部与边界反射情形，同时处理了固定和周期两类边界条件。
Improved Morse Index Stability for Sequences of Harmonic Maps from Degenerating Riemann Surfaces
研究退化黎曼面上调和映射序列的扩展Morse指数稳定性，分析能量二阶变分并确定扩展Morse指数上半连续的条件，改进了早期结果并明确指出退化颈区对极限扩展指数的贡献。
Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains an embedded minimal RP^2. In particular, we derive a generic finiteness result whenever N does not contain a copy of RP^3 in its prime decomposition. We discuss the obstructions to any further generalization of such a result. When the metric g is required to be (scalar positive and) strongly bumpy (meaning that all closed, immersed minimal surfaces do not ha…
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