Thomas Gauglhofer, Hugo Parlier
On a hyperbolic Riemann surface, given two simple closed geodesics that intersect $n$ times, we address the question of a sharp lower bound $L_n$ on the length attained by the longest of the two geodesics. We show the existence of a surface $S_n$ on which there exists two simple closed geodesics of length $L_n$ intersecting $n$ times and explicitly find $L_n$ for $n\leq 3$.
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From curve shortening to flat link stability and Birkhoff sections of geodesic flows
本文利用曲线缩短流证明了闭黎曼曲面测地流的三个新结果：平坦链环在 $C^0$-小扰动下的稳定性、闭测地线的强制存在性定理，以及测地流 Birkhoff 截面的存在性。
For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of $k$ Clifford tori intersecting with maximal symmetry along these two circles. Near each of the circles, after rescaling, the sequences converge smoothly on compact subsets to a Karcher-Scherk tower of order $k$. Sequences of the second type desingularize a collection of the same $k$ Clifford tori supplemented by an…
Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on a random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders prime non-intersecting geodesics by length $γ_1\le γ_2\le ... \le γ_i ,...$, then for fixed $k$, if one allows the genus to go to infinity, the length of $γ_{k}$ is inde…
We show a complete minimal immersion cannot have a certain kind of epitrochoid as a geodesic.
In hyperbolic space $H^n$ we set a geodesic ball of radius $ρ$. Consider a $k$ dimensional minimal submanifold passing through the origin of the geodesic ball with boundary lies on the boundary of that geodesic ball. We prove that its area is no less than the totally geodesic $k$ dimensional submanifold passing through the origin in that geodesic ball. This is a partial generalization of the corresponding problem in $R^n$.
In this survey article we gather classical as well as recent results on minimal geodesics of Riemannian or Finsler metrics, giving special attention to the two-dimensional case. Moreover, we present open problems together with some first ideas as to the solutions.
Minimal hypersurfaces in $\mathbb{S}^5$ with constant scalar curvature and zero Gauss curvature are totally geodesic
本文证明五维球面中常数量曲率且零高斯曲率的闭极小超曲面必为全测地超曲面。
We extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere. This follows work of Brendle and Fraser-Schoen in the euclidean case.
We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of variations and classical analysis only. The geodesic distance is discussed for some special cases and applications towards the theory of nonlinear elasticity are indicated.
In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.
A quadratic lower bound for the number of minimal geodesics
本文改进了Bangert的方法，利用稳定范数单位球，证明了闭黎曼流形上几何上不同的极小测地线数量存在关于第一贝蒂数的二次下界。
Free boundary minimal annuli in geodesic balls of $\mathbb{H}^3$
该论文在双曲三维空间的测地球中构造了可数个单参数族的非旋转自由边界极小环面，这些环面共享棱柱对称群，并作为自由边界双曲悬链面的分叉出现。
Let (T^2, g) be a two-dimensional Riemannian torus. In this paper we prove that the topological entropy of the geodesic flow restricted to the set of initial conditions of minimal geodesics vanishes, independent of the choice of the Riemannian metric.
Hedlund constructed Riemannian metrics on n-tori, $n \geq 3$ for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics are optimal for nilpotent fundamental groups.
On uniqueness of free boundary minimal annuli in geodesic balls of $\mathbb{S}^3_+$ and $\mathbb{H}^3$
本文研究了球面S³₊和双曲空间H³中测地球内自由边界极小环的唯一性，证明其与临界旋转环全等。
On a Riemannian 2-torus $(T^2,g)$ we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all $r \in \mathbb{R} \cup \{\infty\}$ the universal cover $\Br^2$ is foliated by minimal geodesics of rotation number $r$. For irrational $r \in \mathbb{R}$ all geodesics are minimal, for rational $r \in \mathbb{R} \cup \{\i…
The existence of at least two homogeneous geodesics in any homogeneous Finsler manifold was proved in a previous paper by the author. The examples of solvable Lie groups with invariant Finsler metric which admit just two homogeneous geodesics were presented in another paper. In the present work, it is shown that a homogeneous Finsler manifold with indefinite Killing form admits at least four homogeneous geodesics. Examples of invariant Randers metrics on Lie groups with definite Killing form admitting just two homogeneous geodesics and examples with indefinite Killing form admitting just four …
We study the geodesic flow on the normal line congruence of a minimal surface in ${\Bbb{R}}^3$ induced by the neutral Kähler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is shown to be completely integrable. In addition, we give a new holomorphic description of minimal surfaces in ${\Bbb{R}}^3$ and relate it to the classical Weierstrass representation.
Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological entropy of the minimal geodesics coincides with the volume entropy of (M, g) generalizing work of Freire and Mane.
If $M$ is a finite volume complete hyperbolic $3$-manifold, the quantity $\mathcal A_1(M)$ is defined as the infimum of the areas of closed minimal surfaces in $M$. In this paper we study the continuity property of the functional $\mathcal A_1$ with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if $\mathcal A_1(M)$ is realized by a minimal surface satisfying some hypotheses. Understanding the interaction between minimal surfaces and short geodesics in $M$ is the main theme of this paper
We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and prove that within this class the Hopf vector field is a unique global one with totally geodesic property. For the wider class of geodesic unit vector fields on a sphere we give a new necessary and sufficient condition to generate a totally geodesic submanifold in $T_1S^n$.
Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces
研究模2同调中面积最小化子流形的光滑性，证明其一般非光滑（除特定情况），解决怀特猜想，还建立奇异集豪斯多夫维数下界，并证明$\rpt$维罗内塞极小嵌入上的锥是模2面积最小化的。
In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into finitely many real analytic submanifolds called strata. So any component of a shortest-length curve which lies completely in one such stratum is a geodesic in the stratum, hence an embedded real analytic curve. The key question thus is whether there are only finitely many components. F. Albrecht and I.D. Berg proved this is true for a geodesic in a closed regi…
Geodesic orbit pseudo-Riemannian H-type nilmanifolds: case of minimal admissible Clifford modules
本文研究了伪黎曼H型nilmanifold的测地轨道性质，扩展了Riehm关于黎曼H型李群的研究，并针对由最小维度可允许克利福德模块构造的李代数提供了完整特征化。
In real space forms, Fraser and Schoen proved that a free-boundary minimal disk in a geodesic ball is totally geodesic. In this note, we consider free-boundary minimal surfaces $Σ$ (of any genus) in geodesic balls of complex space forms. In $\mathbb{CP}^2$, $\mathbb{C}^2$ and $\mathbb{CH}^2$, we show that if $Σ$ is Lagrangian, then $Σ$ is totally geodesic. In $\mathbb{CP}^n$, $\mathbb{C}^n$ and $\mathbb{CH}^n$ for $n \geq 2$, we show that if $Σ$ has Kähler angle $π/2$, then $Σ$ is superminimal.
This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer $k$, we consider the set of closed geodesics that self-intersect at least $k$ times, and investigate those of minimal length. The main result is that, if the surface has at least one cusp, their self-intersection numbers are exactly $k$ for large enough $k$.
Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes whose leaves are (lifts of) minimal surfaces in $(M,g_t)$. These foliations are deformations of the foliation of $Gr_2(M)$ by (lifts of) totally geodesic planes projected down from the universal cover $\mathbb{H}^3$. Our construction continues to work as long as the sum of the squares of the principal curvatures of the (projections to $M$) of the leaves remains pointwise small…
Closed geodesics with prescribed intersection numbers
本文研究负曲率闭曲面上与给定不相交闭测地线具有指定相交数的素闭测地线计数，通过引入动力学散射算子并利用Pollicott-Ruelle共振理论，给出了其长度趋于无穷时的渐近增长。
We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a simple closed geodesic and a minimal subsphere Sigma^5, which can be equivariantly identified with the Brieskorn sphere W^5_3. As applications we in particular determine the full isometry group of Sigma^7, classify all closed subgroups that act freely, determine the homotopy type of the corresponding orbit spaces, identify the Hirsch-Milnor involution in dimension 5 with the Calabi involution of W^5_3, and obtain explicit formulas for diffeomorphisms between the Brieskorn spheres W^5_3 and W^13_3 with standard Euclidean sp…
We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.
Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the self-intersecting minimal surface that fills the trefoil knot.
On free boundary minimal submanifolds in geodesic balls in $\mathbb H^n$ and $\mathbb S^n_+$
本文在双曲空间和上半球面的测地球中研究自由边界极小子流形，定义了两个自然泛函并证明其临界度量与自由边界极小浸入诱导度量之间的对应关系，同时计算了若干关键例子的 Morse 指标。
Minimal area of Finsler disks with minimizing geodesics
该研究证明内部测地线均为长度极小的Finsler圆盘的Holmes--Thompson面积下界，构造紧例并发现非旋转对称度量可达等号，通过随机测地线离散化度量证明定理，还得到相关积分几何公式在特定Finsler流形上成立的辅助结果。
A well-known Lemma in Riemannian geometry by Klingenberg says that if $x_0$ is a minimum point of the distance function $d(p,\cdot)$ to $p$ in the cut locus $C_p$ of $p$, then either there is a minimal geodesic from $p$ to $x_0$ along which they are conjugate, or there is a geodesic loop at $p$ that smoothly goes through $x_0$. In this paper, we prove that: for any point $q$ and any local minimum point $x_0$ of $F_q(\cdot)=d(p,\cdot)+d(q,\cdot)$ in $C_p$, either $x_0$ is conjugate to $p$ along each minimal geodesic connecting them, or there is a geodesic from $p$ to $q$ passing through $x_0$. …
For an oriented isometric immersion $f:M\to S^n$ the spherical Gauss map is the Legendrian immersion of its unit normal bundle $UM^\perp$ into the unit sphere subbundle of $TS^n$, and the geodesic Gauss map $γ$ projects this into the manifold of oriented geodesics in $S^n$ (the Grassmannian of oriented 2-planes in $\mathbb{R}^{n+1}$), giving a Lagrangian immersion of $UM^\perp$ into a Kaehler-Einstein manifold. We give expressions for the mean curvature vectors for both the spherical and geodesic Gauss maps in terms of the second fundamental form of $f$, and show that when $f$ has conformal sh…
For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cover and that the Busemann functions with these directions are unique up to adding constants. Moreover, using a kind of weak KAM theory, we show that for almost all types of minimal geodesics in the sense of Morse, there is precisely one minimal geodesic of this type.
We prove a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space under some assumptions on the asymptotic behaviour. The result is given in terms of the measure of geodesics intersecting the surface non-trivially, and of a conformal invariant of the curve at infinity.
The existence of two geometrically distinct closed geodesics on an $n$-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27]. We simplify the proof of this statement by the following observation: If for some $N \in \mathbb{N}$ all closed geodesics of index $\le N$ of a non-reversible and bumpy Finsler metric on $S^n$ are geometrically equivalent to the closed geodesic $c$ then there is a covering $c^r$ of minimal index growth, i.e. $${\rm ind}(c^{rm})=m {\rm ind}(c^r)-(m-1)(n-1)$$ for all $m \ge 1$ with ${\rm ind}\…
We prove the existence of minimal surfaces in a bounded convex subset of $\mathbb R^3$, $\mathcal M$, intersecting the boundary of $\mathcal M$ with a fixed contact angle. The proof is based on a min-max construction in the spirit of Almgren-Pitts for the capillarity functional.
We discover new monotonicity formulae for minimal submanifolds in space forms, which imply the sharp area bound for minimal submanifolds through a prescribed point in a geodesic ball. These monotonicity formulae involve an energy-like integral over sets which are, in general, not geodesic balls. In the Euclidean case, these sets reduce to the moving-centre balls introduced by the second author in [Zhu18].
In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following Almgren and Pitts' setting, to produce minimal surfaces with intersecting properties.
Minimal surfaces in the Riemannian product of surfaces
本文研究一般曲面黎曼乘积中的极小曲面，推导全测地曲面的局部形式，证明负曲率因子下无极小2-球面、极小2-环面为拉格朗日的，还给出非零有界曲率下极小2-球面面积的严格下界。
A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $π_1(M,x)$ are contained in a bounded ball, then $π_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $π_1(M,x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $π_1(M,x)$ is virtually abelian.
Froyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in three and four-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal $L$-spaces. In particular, we study relations between Froyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive effective upper bounds for the Froyshov invariants of mini…
We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal surface. In particular, we prove that the only periodic minimal desingularization of a pair of intersecting planes is Scherk's singly-periodic minimal surface.
In this article, based on two case studies, we discuss the role of abnormal geodesics in planar Zermelo navigation problems. Such curves are limit curves of the accessibility set, in the domain where the current is strong. The problem is set in the frame of geometric time optimal control, where the control is the heading angle of the ship and in this context, abnormal curves are shown to separate time minimal curves from time maximal curves and are both small-time minimizing and maximizing. We describe the small-time minimal balls. For bigger time, a cusp singularity can occur in the abnormal …
We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution of Plateau's problem in metric spaces by Lytchak and Wenger as generalizations of minimal surfaces.
New free boundary minimal annuli of revolution in the 3-sphere
本文严格证明3维球面中边界位于测地球面内的自由边界极小环面的存在性，给出嵌入例子及与Otsuki环面的联系，扩展了等周不等式和指标界相关工作至更大测地球。
We prove that a deformation of a hypersurface in a $(n+1)$-dimensional real space form ${\mathbb S}^{n+1}_{p,1}$ induce a Hamiltonian variation of the normal congruence in the space ${\mathbb L}({\mathbb S}^{n+1}_{p,1})$ of oriented geodesics. As an application, we show that every Hamiltonian minimal sumbanifold in ${\mathbb L}({\mathbb S}^{n+1})$ (resp. ${\mathbb L}({\mathbb H}^{n+1})$) with respect to the (para-) Kaehler Einstein structure is locally the normal congruence of a hypersurface $Σ$ in ${\mathbb S}^{n+1}$ (resp. ${\mathbb H}^{n+1}$) that is a critical point of the functional ${\ca…
Existence of Time-like Geodesics in Asymptotically Flat Spacetimes: A Generalized Topological Criterion
本文提出并证明了一个广义拓扑判据，用于判定渐近平坦时空中类时测地线的存在性，并揭示其对时空结构、Lorentzian 度量障碍及 Einstein 约束方程的意义。
On the minimal number of closed geodesics on positively curved Finsler spheres
该研究针对满足特定曲率与可逆性条件的正曲率Finsler球面，证明了其上素闭测地线的最小数量下界，解决了偶数维情形下的Katok-Anosov猜想，并给出了有限闭测地线情形下非双曲闭测地线的数量下界。
Given two points on a soup can or conical cup with lid, we find and classify all paths of minimal length connecting them. When the number of minimal paths is finite, there are at most four on a can and three on a cup. At worst, minimal paths are piece-wise smooth with three components, each of which is a classical geodesic. Minimal paths are geodesics in the sense of Banchoff.
In a seminal paper published in $1968$, J. Simons proved that, for $n\leq 5$, the Euclidean (minimal) cone $CM$, built on a closed, oriented, minimal and non totally geodesic hypersurface $M^n$ of $\mathbb S^{n+1}$ is unstable. In this paper, we extend Simons' analysis to {\em warped} (minimal) cones built over a closed, oriented, minimal hypersurface of a leaf of suitable warped product spaces. Then, we apply our general results to the particular case of the warped product model of the Euclidean sphere, and establish the unstability of $CM$, whenever $2\leq n\leq 14$ and $M^n$ is a closed, or…
We prove that the area of a free boundary minimal surface $Σ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $Σ$ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows work of Brendle and Fraser-Schoen in the euclidean setting.
Some new functionals related to free boundary minimal submanifolds
本文研究了自由边界极小子流形相关的新的泛函，并证明了这些泛函的临界度量与自由边界极小浸入有关。
Generalization of Newton's minimal resistance problem to Riemannian surfaces
将牛顿最小阻力问题推广到具有测地坐标系的黎曼曲面（包括球面和双曲平面），通过沿径向测地线流动的流体粒子推导阻力泛函，证明光滑极值曲线为曲面的斜驶线，并分析约束最小化问题，证明光滑极值曲线不存在强局部极小值，刻画全局极小化子。
The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. Through every point in such a metric there is a geodesic that saturates the length condition, and saturating geodesics in a given homotopy class form a band. The extremal metric is unknown when bands of geodesics cross, as it happens for surfaces of non-zero genus. We use recently proposed convex programs to numerically find the minimal-area metric on the square torus with a square boundary, for …
Singularities of Discrete Indefinite Affine Minimal Surfaces
本文研究离散不定仿射极小曲面的奇点，通过离散Lelieuvre公式定义离散渐近网的奇异边和顶点，使光滑情形中奇异集的关键性质在离散情形下仍然保持。
An embedded cubic graph consisting of segments of geodesics such that the angles at any vertex are equal to $2π/3$ is a closed local minimal net. This net is regular if all segments of geodesics are equal. The problem of classification of closed local minimal nets on surfaces of constant negative curvature has been formulated in the context of the famous Plateau problem in the one-dimensional case. In this paper we prove an asymptotic for $\sharp (W^r(g))$ as $g\to +\infty$ where $g$ is genus and $W^r(g)$ is the set of the regular single-face minimal nets on surfaces of curvature -1. Then we c…
In contrast with the 3-dimensional case (cf. \cite{RaMo}), where rotationally symmetric totally geodesic free boundary minimal surfaces have Morse index one; we prove in this work that the Morse index of a free boundary rotationally symmetric totally geodesic hypersurface of the $n$-dimensional Riemannnian Schwarzschild space with respect to variations that are tangential along the horizon is zero, for $n\geq4$. Moreover, we show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, $n\geq 8$, with Morse index equal to $0$. Also, it is shown that, for…
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