Lower bounds for index of Wente tori

A note on the index of closed minimal hypersurfaces of flat tori

Generalizing earlier work by Ros in ambient dimension three, we prove an affine lower bound for the Morse index of closed minimal hypersurfaces inside a flat torus in terms of their first Betti number (with purely dimensional coefficients).

Lower bounds for Morse index of constant mean curvature tori

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.

Computation of the index of some meromorphic functions of degree 3 on tori

The index of a meromorphic function $g$ on a compact Riemann surface is an invariant of $g$, which is defined as the number of negative eigenvalues of the differential operator $L:=-Δ-|dG|^2$, where $Δ$ is the Laplacian with respect to a conformal metric $ds^2$ on the Riemann surface, $G \colon M \to S^2$ is the holomorphic map corresponding to $g$. We consider the meromorphic function $w$ on the Riemann surface $M_a= \{(z,w) \in\widehat{\mathbb{C}}^2 \mid w^2=z(z-a)(z+\frac{1}{a})\}(a \geqslant 1 )$ homeomorphic to a torus, and we determine the index of $tw$ for all $a$ in the range $1 \leqsl…

关于4维球面中极小2维环面的指数

On the index of minimal 2-tori in the 4-sphere

本文研究球面中极小2维环面的Morse指数问题，证明了4维球面中极小2维环面指数下界为6及等号成立条件，还给出Hopf微分消失的n维球面中极小2维环面的指数下界与紧例。

Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori

A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using the index of the self-shrinker when viewed as a critical point of the entropy functional. In this work, we prove an upper bound on the index of rotationally symmetric self-shrinking tori in terms o…

Otsuki 环面的双极曲面的指标

Index of bipolar surfaces to Otsuki tori

本文研究了 Otsuki 环面经 Lawson 双极曲面构造法得到的 $\mathbb S^4$ 中极小环面，给出了当参数接近 $\sqrt 2/2$ 时其 Morse 指标与零化数的上下界，并提出了数值辅助猜想。

CMC tori of revolution in $\mathbb{S}^3$: additional data on the spectra of their Jacobi operators

We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions. These theorems provide accurate numerical methods for finding the spectra of those operators over either type of function space. As an application, we numerically compute the Morse index of constant mean curvature tori of revolution in the unit 3-sphere $\mathbb{S}^3$, confirming that every such torus has Morse index at least five, and showing that other kno…

环面上的$L^\infty$-度量与Schoen猜想

$L^\infty$-metrics on tori and Schoen's conjecture

本文在奇异集基本群非满射的附加假设下，证明了环面上$L^\infty$-度量的Schoen猜想，即度量可延拓为光滑平坦度量。

Stable CMC and index one minimal surfaces in conformally flat manifolds

Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $Σ$ be a compact connected and orientable surface immersed in $M$ which is a stable constant mean curvature (CMC) surface or an index one minimal surface. We prove that $Σ$ is homeomorphic either to a sphere or to a torus. Moreover, in case $Σ$ is homeomorphic to a torus, then it is embedded, minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unit …

Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces

The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain modu…

Metrics without Morse index bounds

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.

椭圆泛函的古代梯度流与Morse指数

Ancient gradient flows of elliptic functionals and Morse index

本文研究黎曼流形中椭圆泛函梯度流的闭古代解，对球面中平均曲率流及曲线缩短流在特定面积或长度界下进行了完全分类，揭示了其解仅为稳态或收缩的赤道球面、Clifford环面与圆。