Mostafa Esfahani Zadeh
In this paper we state and prove Morse type inequalities for Morse functions as well as for closed differential 1-forms. These inequalities involve delocalized Betti numbers. As an immediate consequence, we prove the vanishing of delocalized Betti numbers of manifolds fibering over the circle.
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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…
We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannian manifold and $Σ\subset M$ be a closed embedded minimal hypersurface with area at most $A>0$ and with a singular set of Hausdorff dimension at most $n-7$. We show the following bounds: there is $C_A>0$ depending only on $n$, $g$, and $A$ so that $$\sum_{i=0}^n b^i(Σ) \leq C_A \big(1+index(Σ)\big) \quad \text{ if $3\leq n+1\leq 7$},$$ $$\mathcal{H}^{n-7}\big(Sing(Σ)\big) \leq C_A \big(1+index(Σ)\big)^{7/n} \quad \text{ if $n+1\geq 8$},$$ where…
We generalize a method by L. Ambrozio, A. Carlotto, and B. Sharp to study the Morse index of closed f-minimal hypersurfaces isometrically immersed in a general weighted manifold. The technique permits, in particular, to obtain a linear lower bound on the Morse index via the first Betti number for closed f-minimal hypersurfaces in products of some compact rank one symmetric spaces with an Euclidean factor, endowed with the rigid shrinking gradient Ricci soliton structure. These include, as particular cases, all cylindric shrinking gradient Ricci solitons.
We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their variation with this parameter.
We use Morse theory of the Yang-Mills functional to compute the Betti numbers of the moduli stack of flat U(3)-bundles over a compact nonorientable surface. Our result establishes the antiperfection conjecture of Ho-Liu, and provides evidence for the equivariant formality conjecture of the author.
Index estimate by first Betti number of minimal hypersurfaces in compact symmetric spaces
本文证明紧致半单对称空间中不稳定闭极小超曲面的Morse指标被其第一Betti数的常数倍下界控制，方法为先前方法的自然推广，提供指标估计新途径。
Morse inequalities for noncompact manifolds
本文在离散群余紧真不连续作用于非紧流形的条件下，利用 $L^2$-贝蒂数及描述临界点粗略构型的群上函数，建立了莫尔斯函数不要求群不变时的莫尔斯不等式。
We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In the complete non-compact case, the lower bound is in terms of the dimension of the space of weighted square summable $f$-harmonic $1$-forms; in particular, in dimension $2$, the procedure gives an in…
Ricci curvature and minimal hypersurfaces with large Betti numbers
研究构造了高维里齐曲率正的闭合流形，其嵌入的双侧极小超曲面具有Morse指数为一且贝蒤数不一致有界的特性。
It is well known that the cohomology groups of a closed manifold $M$ can be reconstructed using the gradient dynamical of a Morse-Smale function $f\colon M\to \R$. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of $f$ in term of Betti numbers of $M$. These inequalities can be deduced through a purely analytic method by studying the asymptotic behaviour of the deformed Laplacian operator. This method was introduced by E. Witten and has inspired a numbers of great achievements in Geometry and Topology in few past decades. I…
Dynamical Morse entropy
本文针对可平铺顺从群作用下的拓扑空间连续函数，定义了同调可检测临界点数量的熵，证明其下界为贝蒂数熵之和，推广了微分几何中的标准莫尔斯不等式。
Hyperbolic 4-manifolds with perfect circle-valued Morse functions
本文构造了具有完美圆值Morse函数的有限体积双曲4-流形，证明了存在无限多个此类流形，其具有有界数量的1-和3-把手分解及有界的Betti数。
For a Morse function f on a compact oriented manifold M, we show that f has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in M whose components have nontrivial linking number, such that the minimal value of f on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of f in terms of the Betti numbers of M and the behavior of f with respect to links. This can be viewed as a refinement, in the case of compact manifolds, of the Rabinowitz Saddle …
For a Morse function on a closed orientable Riemannian manifold one introduces the {\it virtually small spectral package} an analytic object consisting of a finite number of analytic quantities derived from the pair, {\it Riemannian metric, Morse function\} which, in principle, can be calculated. One shows that they determine the {\it Torsion } of the underlying space, a parallel to the result that the dimensions of the spaces of harmonic forms calculate the {\it Euler-Poincaré characteristic} of the underlying space and extends the {\it Poincaré Duality} between harmonic forms and between Bet…
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.
Let F be a flat vector bundle over a compact Riemannian manifold M and let f be a Morse function. Let g be a smooth Euclidean metric on F, let g_t=e^{-2tf}g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g_t. Assuming that the vector field grad(f) satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for \log(ρ(t)) for t\to +\infty of the form a_0+a_1t+b\log\left(\frac tπ\right)+o(1), where the coefficient b is a half-integer depending only on the Betti numbers of F. In the case where all the critical values of f are rational, we…
In this short note, we compute the Betti numbers of the moduli stack of flat SU(3)-bundles over a Klein bottle. We also handle the general compact group case over RP^2. In all cases the cohomology is found to be equivariantly formal, supporting a conjecture from the author's doctoral thesis. Our results also verify conjectural formulas obtained by Ho-Liu using Yang-Mills Morse theory.
In this paper, we consider immersed two-sided minimal hypersurfaces in $\mathbb{R}^n$ with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When $n=4$, we are able to drop the nullity term by a careful study for the rigidity case. Our result is the first effective generalization of Li-Wang. Using our index estimates and ideas from the recent work of Chodosh-Ketover-Maximo, we prove compactness and finiteness results of minimal h…
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