Isabel M. C. Salavessa
By only using spectral theory of the Laplace operator on spheres, we prove that the unit 3-dimensional sphere of a 2-dimensional complex subspace of $\mathbb{C}^3$ is a $Ω$-stable submanifold with parallel mean curvature, when $Ω$ is the Kähler calibration of rank 4 of $\mathbb{C}^3$.
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In the present study we consider knotted spheres in Euclidean $4$-space $ \mathbb{E}^{4}$. Firstly, we give some basic curvature properties of knotted spheres in $ \mathbb{E}^{4}$. Further, we obtained some results related with the conjugate nets and Laplace transforms of these kind of surfaces.
In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mean curvature surfaces in S^3(a), 1/3 <= a < 1 proving that they are spheres or the minimal Clifford torus in S^3(1/3). This allows to solve the isoperimetric problem in these Berger spheres.
The aim of the paper is to investigate the rigidity and the deformability of pseudoholomorphic curves in the nearly K{ä}hler sphere $\mathbb{S}^6,$ among minimal surfaces in spheres. Under various assumptions we describe the moduli space of all noncongruent minimal surfaces $f\colon M\to\mathbb{S}^n$ that are isometric to a pseudoholomorphic curve in $\mathbb{S}^6.$ Moreover, we prove a Schur type theorem (see \cite[p. 36]{Chnew}) for minimal surfaces in spheres.
A theorem by Almgren establishes that any minimal $2$-sphere immersed in $\mathbb{S}^3$ is a totally geodesic equator. In this paper we give a purely geometric extension of Almgren's result, by showing that any immersed, real analytic $2$-sphere in $\mathbb{S}^3$ that is saddle, i.e., of non-positive extrinsic curvature, must be an equator of $\mathbb{S}^3$. We remark that, contrary to Almgren's theorem, no geometric PDE is imposed on the surface. The result is not true for $C^{\infty}$ spheres.
A correspondence is established between a class of minimal immersed surfaces of $\mathbb{S}^3(2)$ and area-minimizing unit vector fields defined on the antipodally punctured unit sphere $\mathbb{S}^2\backslash\{N,S\}$. As a consequence, we establish a stability relation for Lawson cylinders in $\mathbb{S}^3(2)$.
Spheres with parallel mean curvature in $\mathbb{S}^2 \times \mathbb{H}^2$
本文研究球面与双曲平面乘积空间中平行平均曲率曲面，分类了因拓扑原因使全纯二次微分消失的球面。
In this paper we first prove a characterization formula for biharmonic maps in Euclidean spheres and, as an application, we construct a family of biharmonic maps from a flat $2$-dimensional torus $\mathbb{T}$ into the $3$-dimensional unit Euclidean sphere $\mathbb{S}^3$. Then, for the special case of maps between spheres whose components are given by homogeneous polynomials of the same degree, we find a more specific form for their bitension field. Further, we apply this formula to the case when the degree is $2$, and we obtain the classification of all proper biharmonic quadratic forms from $…
Let $M$ be a smooth, connected, compact submanifold of $\mathbb{R}^n$ without boundary and of dimension $k\geq 2$. Let $\mathbb{S}^k \subset \mathbb{R}^{k+1}\subset \mathbb{R}^n$ denote the $k$-dimesnional unit sphere. We show if $M$ has reach equal to one, then its volume satisfies $\text{vol}(M)\geq \text{vol}(\mathbb{S}^k)$ with equality holding only if $M$ is congruent to $\mathbb{S}^k$.
In this paper, we partially solve Yau' Conjecture of the first eigenvalue of an embedded compact minimal hypersurface of unit sphere $\mathbb{S}^{n+1}(1)$, i.e., Corollary 1.2. In particular, Corollary 1.3 proves that the condition $\int_{Ω_{1}}|\nabla u|^{2}=(n+1)\int_{Ω_{1}}u^{2}$ is naturally true and meaningful in Corollary 1.2.
In \cite{Zh} \cite{LY} Zhang, Luo and Yin initiated the study of Lagrangian submanifolds satisfying ${\rm \nabla^*} T=0$ or ${\rm \nabla^*\nabla^*}T=0$ in $\mathbb{C}^n$ or $\mathbb{CP}^n$, where $T ={\rm \nabla^*}\tilde{h}$ and $\tilde{h}$ is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ${\rm \nabla^*} T=0$ or ${\rm \nabla^*\nabla^*}T=0$ in $\mathbb{C}^2$ under proper small energy assumption and gave new characterization of the Whitney spheres in $\mathbb{C}^2$. In this paper we extend these results to Lagrangian s…
We construct examples of spherical space forms $(S^3/Γ,g)$ with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at $(S^3/Γ,g)$: a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in dimension $3$ a general contruction of Type I neckpinching and clarify the relationship between stable spheres and non-trivial Type I singularities of the Ricci flow. Some symmetry assumptions prevent the appearance of stable spheres, and this has consequences on the types of sing…
Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable compact minimal surfaces of the product of a 2-sphere and any Riemann surface are the complex ones.
On Stability and Isoperimetry of Constant Mean Curvature Spheres of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R.$
本文研究了 $\mathbb H^n\times\mathbb R$ 和 $\mathbb S^n\times\mathbb R$ 中旋转常平均曲率球的稳定性与等周性，证明了特定条件下球的稳定性，确立了等周问题解的唯一性，并指出并非所有稳定球都是等周的。
Stable Abelian Yang--Mills--Higgs Fields on the Round Three-Sphere
本文证明了三维球面上自对偶阿贝尔杨-米尔斯-希格斯能量的稳定弱临界点必为真空，并给出非稳定临界点的Hessian最小特征值上界。
We introduce a class of minimal submanfolds $M^n$, $n\geq 3$, in spheres $\mathbb{S}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of equally ruled minimal isometric deformations that are genuine. As for compact examples, there are plenty of them but only for dimensions $n=3$ and $n=4$. In the first case, we have that $M^3$ must be a $\mathbb{S}^1$-bundle over a minimal torus $T^2$ in $\mathbb{S}^5$ and in the second case $M^4$ has to be a $\mathbb{S}^2$-bundle over a minimal sphere $\mathb…
On the curvature bounded sphere problem in $\mathbb{R}^3$
该研究证明了在$\mathbb{R}^3$中，满足法曲率有界条件的拓扑球体若被包含在半径为2的开球内，则其包围区域必须包含单位球，并探讨了相关体积问题。
SubRiemannian geometry on the sphere $\mathbb{S}^3$
本文首先介绍了四元数的概念，然后定义三维球面为长度为1的四元数的集合。四元数群在$\mathbb{S}^3$上诱导出非交换李群的结构。这个群是紧的，在这种情况下得到的结果与在海森堡群的情况下得到的结果非常不同，海森堡群是一个非紧的李群。像海森堡群的情况下，我们引入了一个不可积分布的领域和度量它使用两个非交换的左不变向量场。这样，$\mathbb{S}^3$就成为一个次黎曼流形。已知群$SU(2) $与球面$\mathbb{S}^3$同构，并表示其中元素为矩阵的次黎曼流形的一个例子。本文的主要问题是研究这类流形上的水平曲线连通性及其测地线。本文利用拉格朗日方法研究了${\mathbb S}^3$上具有最小弧长的水平曲线的连通性定理。我们证明了对于${\mathbb S}^3$中的任意两点，存在一条连接这两点的测地线。
We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmuş-Montaldo-Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces $\mathbb S^{n+1}$ are either the hypersphere $\mathbb S^n(1/\sqrt2)$ or the Clifford hypersurface $\mathbb S^{n_1}(1/\sqrt2)\times\mathbb S^{n_2}(1/\sqrt2)$ with $n_1+n_2=n$ and $n_1\neq n_2$. Moreover, we also show that there does not exi…
In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\mathbb{S}^{n+d}$.
We deal with minimal surfaces in spheres that are locally isometric to a pseudoholomorphic curve in a totally geodesic $\mathbb{S}^{5}$ in the nearly K{ä}hler sphere $\mathbb{S}^6$. Being locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5$ turns out to be equivalent to the Ricci-like condition $Δ\log(1-K)=6K,$ where $K$ is the Gaussian curvature of the induced metric. Besides flat minimal surfaces in spheres, direct sums of surfaces in the associated family of pseudoholomorphic curves in $\mathbb{S}^5$ do satisfy this Ricci-like condition. Surfaces in both classes are exceptional …
A characterisation of the Calabi product of hyperbolic affine spheres
存在一种公知的构造，其允许将两个双曲仿射球$f_i: M_i^{n_i} \to \mathbb R^{n_i+1}$与$I \times M_1 \times M_2$到$\mathbb R^{n_1+n_2+3}$中的新的双曲仿射球浸入相关联。在本文中，我们处理的反问题：如何确定的性质的差张量是否一个给定的双曲仿射球浸入的流形$M^n \to \mathbb R^{n+1}$可以分解的方式。
We consider the sub-Riemannian $3$-sphere $(\mathbb{S}^3,g_h)$ obtained by restriction of the Riemannian metric of constant curvature $1$ to the planar distribution orthogonal to the vertical Hopf vector field. It is known that $(\mathbb{S}^3,g_h)$ contains a family of spherical surfaces $\{\mathcal{S}_λ\}_{λ\geq 0}$ with constant mean curvature $λ$. In this work we first prove that the two closed half-spheres of $\mathcal{S}_0$ with boundary $C_0=\{0\}\times\mathbb{S}^1$ minimize the sub-Riemannian area among compact $C^1$ surfaces with the same boundary. We also see that the only $C^2$ solut…
Ruled surfaces and hyper-dual tangent sphere bundle
本文研究超对偶向量中的单位超对偶球与 E-Study 映射，证明相应球面与切丛同构，并建立直纹面可展条件、曲线几何解释及其与超对偶曲线的关系。
In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices modulo an appropriate group action. Using this description, many examples, such as constantly curved holomorphic 2-spheres of higher degree, nonhomogenous minimal 2-spheres of constant curvature, etc., are constructed. Uniqueness is proven for the totally real constantly curved …
A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann-Cartan space is introduced in a natural way. For the case of the odd dimensional spheres $\mathbb{S}^{2n+1}$ viewed as homogeneous spaces of the special unitary groups, the classical Nomizu's Theorem on invariant connections has permitted to obtain an algebraical description of all the connections which turn the sphe…
Embedded constant mean curvature hypertori in the $2n$-sphere
该研究在$2n$维标准球面中构造出两个拓扑类型为$\mathbb{S}^{n-1} \times \mathbb{S}^{n-1} \times \mathbb{S}^1$、具有相同负平均曲率的不同嵌入常平均曲率超环面，拓展了球面极小/常平均曲率嵌入超曲面的相关研究。
In this paper, we study biconservative surfaces with parallel normalized mean curvature vector field ($PNMC$) in the $4$-dimensional unit Euclidean sphere $\mathbb{S}^4$. First, we study the existence and uniqueness of such surfaces. We obtain that there exists a $2$-parameter family of non-isometric abstract surfaces that admit a (unique) $PNMC$ biconservative immersion in $\mathbb{S}^4$. Then, we obtain the local parametrization of these surfaces in the $5$-dimensional Euclidean space $\mathbb{E}^5$.
In this paper, we prove that minimal hypersurfaces when $n\geq 3$ and nonzero constant mean curvature hypersurfaces when $n\geq2$ foliated by spheres in parallel horizontal hyperplanes in ${\mathbb{H}}^n \times \mathbb{R}$ must be rotationally symmetric.
Bryant \cite{Bryant84} classified all Willmore spheres in $3$-space to be given by minimal surfaces in $\mathbb R^3$ with embedded planar ends. This note provides new explicit formulas for genus 0 minimal surfaces in $\mathbb R^3$ with $2k+1$ embedded planar ends for all $k\geq4.$ Peng and Xiao claimed these examples to exist in \cite{PengXiao2000}, but in the same paper they also claimed the existence of a minimal surface with 7 embedded planar ends, which was falsified by Bryant \cite{Bryant88}.
We study generalized Killing spinors on round spheres $\mathbb{S}^n$. We show that on the standard sphere $\mathbb{S}^8$ any generalized Killing spinor has to be an ordinary Killing spinor. Moreover we classify generalized Killing spinors on $\mathbb{S}^n$ whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola--Friedrich's canonical spinor on 3-Sasakian manifolds of dimension 7. Finally we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.
For a singular Riemannian foliation $\mathcal{F}$ on a Riemannian manifold, a curve is called horizontal if it meets the leaves of $\mathcal{F}$ perpendicularly. For a singular Riemannian foliation $\mathcal{F}$ on a unit sphere $\mathbb{S}^{n}$, we show that if $\mathcal{F}$ is a polar foliation or if $\mathcal{F}$ is given by the orbits of an infinitesimally polar action, then the horizontal diameter of $\mathbb{S}^{n}$ is $π$, i.e., any two points in $\mathbb{S}^{n}$ can be connected by a horizontal curve of length $\leqπ$.
The stability and the index of compact minimal submanifolds of the Berger spheres $\mathbb{S}^{2n+1}_τ,\, 0<τ\leq 1$, are studied. Unlike the case of the standard sphere ($τ=1$), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if $τ^2\leq 1/2$. Moreover, there are no stable compact minimal $d$-dimensional submanifolds of $\mathbb{S}^{2n+1}_τ$ when $1 / (d+1) < τ^2 \leq 1$ and the stable ones are classified for $τ^2=1 / (d+1)$ when the submanifold is embedded. Finally, the compact orientable minimal surfaces of $\mathbb{S}^3_τ$ with index …
Quantitative $C^1$-stability of spheres in rank one symmetric spaces of non-compact type
本文证明在非紧型秩一对称空间中，测地球面在微小 $C^1$-保体积扰动下具有一致的定量稳定性，利用 Laplacian 显式谱隙量化周长增加，并定量证明了小体积下测地球面为等周区域。
In this paper we discuss the stability of geodesic spheres in $\mathbb{S}^{n+1}$ under constrained curvature flows. We prove that under some standard assumptions on the speed and weight functions, the spheres are stable under perturbations that preserve a volume type quantity. This extends results by Escher and Simonett, 1998, and the author, 2015, to a Riemannian manifold setting.
We show that if the entropy of any closed hypersurface is close to that of a round hyper-sphere, then it is close to a round sphere in Hausdorff distance. Generalizing the result of \cite{BW1} to higher dimensions.
Let $(M, g)$ be a complete Riemannian $3$-manifold that is asymptotic to Schwarzschild with positive mass and whose scalar curvature vanishes. We \textsl{unconditionally} characterize the large, embedded stable constant mean curvature spheres in $(M, g)$.
On the Inscribed Sphere and Concurrent Lines through the Centers of Apollonius Spheres in $\mathbb{R}^n$
本文研究了在R^n中由共同家族球体构成的阿波罗尼奥斯问题解之间的关系，证明了通过解中心的直线交于一点，并推广了Morita的定理到n维空间。
In this manuscript we study rotationally $p$-harmonic maps between spheres. We prove that for $p\in\mathbb{N}$ given, there exist infinitely many $p$-harmonic self-maps of $\mathbb{S}^m$ for each $m\in\mathbb{N}$ with $p<m< 2+p+2\sqrt{p}$. In the case of the identity map of $\mathbb{S}^m$ we explicitly determine the spectrum of the corresponding Jacobi operator and show that for $p>m$, the identity map of $\mathbb{S}^m$ is stable when interpreted as a $p$-harmonic self-map of $\mathbb{S}^m$.
The author proves that there is an open non empty set of metrics on any 3-manifold such that there exists a family of stably embedded minimal 2-spheres whose area is unbounded. This generalizes the work of T. Colding and W. Minicozzi who have shown an analogous result for the torus and B. Dean who showed the positive genus case.
Optimal regularity of stable harmonic maps to spheres
本文研究到球面的稳定调和映射的正则性，确定其奇异集余维数的下界，证明结果最优，并给出n小于k时n维到k维球面非常数调和映射的非平凡指标界。
For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ with $n \geq 3$, in terms of what we have called {\it generalised complex orthogonal Stiefel manifolds} of $\mathbb{C}^{n+1}$. In particular, the twistor space $\mathfrak{Z}(\mathbb{S}^{6})$ is biholomorphic…
Stable proper biharmonic maps in Euclidean spheres
本文构造了从单位球到欧几里得球面的首个显式稳定真弱双调和映射族，通过建立双能量第二变分公式并分析特定映射的稳定性实现。
The loop space $L\mathbb{P}_1$ of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to $\mathbb{P}_1$ is an infinite dimensional complex manifold. The loop group $LPGL(2,\mathbb{C})$ acts on $L\mathbb{P}_1$ . We prove that the group of $LPGL(2,\mathbb{C})$ invariant holomorphic line bundles on $L\mathbb{P}_1$ is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of these bundles is finite dimensional, and compute the dimension for a generic bundle.
We give examples of asymptotically flat three-manifolds $(M,g)$ which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold. This resolves a question raised by G. Huisken and S.-T. Yau in 1996. On the other hand, we show that such surfaces cannot exist when $(M,g)$ has nonnegative scalar curvature. This result depends on an intricate relationship between the scalar curvature of the initial data set and the isoperimetric ratio of large stable constant mean curvature surfaces.
The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow converges in the Gromov-Hausdorff topology to a limiting metric space which is also a 2-sphere, but with different marked points and hence a different complex structure. The limiting metric space carries a unique conical constant curvature metric in the semi-stable case, and a un…
We construct Bing houses in all dimensions $n \geq 3$, obtaining non-separating PL immersions of $\mathbb{S}^n \rightarrow\mathbb{R}^{n+1}$.
It's well known that the n-sphere $S^n$ is the universal double covering of the $n$-dimensional real projective space $\mathbb{R}P^n$ and then any Finsler metric on $\mathbb{R}P^n$ induces a Finsler metric of $S^n$. In this paper, we prove that for every Finsler $(S^n, F)$ for $n\geq3$ whose metric is induced by irreversible Finsler $(\mathbb{R}P^n,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $(\fracλ{λ+1})^2<K\leq 1$, there exist at least $n-1$ prime closed geodesics on $(S^n, F)$. Furthermore, if there exist finitely many distinct closed geodesics on $(S^n, F)$, then there ex…
By comparing the Laplace spectrum of the sphere $\mathbb{S}^n$ to its Weyl function $w(x) = \frac{ω_n}{(2π)^n}|\mathbb{S}^n|x^{n/2}$, we show that no analogue of Pólya's eigenvalue conjecture holds in general for Riemannian manifolds with positive sectional curvature.
We prove that an integral Cauchy-Riemann inequality holds for any pair of smooth functions $(f,h)$ on the 2-sphere $\mathbb{S}^2$, and equality holds iff $f$ and $h$ are related $λ_1$-eigenfunctions. We extend such inequality to 4-tuples of functions, only valid on the $L^2$-orthogonal complement of a suitable nonzero finite dimensional space of functions. As a consequence we prove that 2-spheres are not $Ω$-stable surfaces with parallel mean curvature in $\mathbb{R}^7$ for the associative calibration $Ω$.
Equivariant constructions of spheres with Zoll families of minimal spheres
本文通过等变方法构造了具有Zoll家族极小球的球体，并展示了其在不同维度中的应用。
Let $Σ$ be an oriented compact hypersurface in the round sphere $\mathbb{S}^n$ or in the flat torus $\mathbb{T}^n$, $n\geq 3$. In the case of the torus, $Σ$ is further assumed to be contained in a contractible subset of $\mathbb{T}^n$. We show that for any sufficiently large enough odd integer $N$ there exists an eigenfunctions $ψ$ of the Laplacian on $\mathbb{S}^n$ or $\mathbb{T}^n$ satisfying $Δψ=-λψ$ (with $λ=N(N+n-1)$ or $N^2$ on $\mathbb{S}^n$ or $\mathbb{T}^n$, respectively), and with a connected component of the nodal set of $ψ$ given by~$Σ$, up to an ambient diffeomorphism.
The Gromov-Hausdorff distance between spheres
该研究给出不同维数圆度量球面间Gromov-Hausdorff距离的通用上下界，借助拓扑思路推导下界，通过构造最优对应证明五类情形下界紧，并提出相关开放问题。
In this paper, we obtain Pizzetti-type formulae on regions of the the unit sphere $\mathbb{S}^{m-1}$ of $\mathbb{R}^m$, and study their applications to the problem of inverting the spherical Radon transform. In particular, we approach integration over $(m-2)$-dimensional sub-spheres of $\mathbb{S}^{m-1}$, $(m-1)$-dimensional sub-balls, and over $(m-1)$-dimensional spherical caps as the action of suitable concentrated delta distributions. In turn, this leads to Pizzetti formulae that express such integrals in terms of the action of SO$(m-1)$-invariant differential operators. In the last section…
Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.
We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded Kasparov cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on $\mathbb R^{n+1}$. We find that the resulting spectral triple for the algebra $C(\mathbb S^n)$ differs from the Dirac operator on the round sphere by a so-called index cycle, whose class in $KK_0(\mathbb C, \mathbb C)$ represents the multiplicative unit. At all points we check that our construction involving the unbounded Kasparov product is compa…
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