Oussama Hijazi, Sebastian Montiel, Xiao Zhang
New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of Atiyah-Patodi-Singer type. Spinorial techniques are used to give simple proofs of classical results for compact embedded hypersurfaces.
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In this study, we obtain a spinorial Gauss formula for a lightlike hypersurface in Lorentzian manifold with 4-dimension. Then, we take into account the changes caused by degenerate metric on hypersurface and investigate Dirac operator for lightlike hypersurface. Later, we establish the relation between Dirac operators and Riemannian curvatures of manifold and hypersurface.
We show that the eigenvalues of the intrinsic Dirac operator on the boundary of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean Dirac operators, either in the domain with a MIT-bag type boundary condition or in the whole space, with a suitably chosen zero order mass term.
In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in $\R^{n}$ (Delaunay unduloids). When $n=3$, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space.
We generalize the transgression formula for the eta form of Bismut, Cheeger and Berline, Getzler, Vergne for vertical Dirac operators on a fibre bundle with odd dimensional fibres where the Dirac operators have locally at most one eigenvalue of multiplicity one crossing zero transversally.
We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the first eigenvalue of a drifting Schrödinger operator on the hypersurface. Moreover, using a recent approach developed by O. Hijazi and S. Montiel, we completely characterize the equality case when the ambient manifold is the standard hyperbolic space.
Lower bound for the first eigenvalue of a minimally embedded hypersurface in a Riemannian manifold
本文研究了Riemannian流形中极小嵌入超曲面的谱性质，在Ricci曲率具有正下界的紧致可定向流形条件下，给出了闭可定向超曲面上Laplace-Beltrami算子第一特征值的下界。
Examples of compact embedded $λ$-hypersurfaces
本文构造了紧致嵌入的$\lambda$-超曲面，其与球面微分同胚但不等距于标准球面，从而否定了Alexandrov型定理对$\lambda$-超曲面的成立。
In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained t…
Examples of compact embedded mean convex $λ$-hypersurfaces
本文研究了 λ>0 时紧凑嵌入的 mean 凸 λ-超曲面，构造了与球面同胚但非标准球面的例子，并证明了 λ>0 时不存在其他与球面同胚的紧凸嵌入 λ-超曲面。
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In this paper we will prove Hadamard-Stoker type theorems in the following ambient spaces: $\man ^n \times \r$, where $\man ^n $ is a $1/4-$pinched manifold, and certain Killing submersions, e.g., Berger spheres and Heisenberg spaces. That is, under the condition that the principal curvatures of an immersed hypersurfaces are greater than some non-negative constant (depending on the ambient space), we prove that such a hypersurface is embedded and we also study its topology.
Let $M$ be a globally hyperbolic manifold with complete spacelike Cauchy hypersurface $Σ\subset M$. Building on past and recent works of Bär and Strohmaier, we extend their Fredholm result of the Atiyah-Singer Dirac operator on compact Lorentzian spaces to the case, where $M$ is diffeomorphic to a product of $Σ$ with a compact time intervall and the hypersurface is a Galois covering with respect to a group $Γ$. We follow the first approach of both authors in this extended setting, where a well-posedness result of the Cauchy problem for the Dirac operator on non-compact manifolds is needed in p…
In this paper we show the existence of a closed, embedded $λ$-hypersurfaces $Σ\subset \mathbb{R}^{2n}$. The hypersurface is diffeomorhic to $\mathbb{S}^{n-1} \times \mathbb{S}^{n-1} \times \mathbb{S}^1$ and exhibits $SO(n) \times SO(n)$ symmetry. Our approach uses a "shooting method" similar to the approach used by McGrath in constructing a generalized self-shrinking torus solution to mean curvature flow. The result generalizes the $λ$-torus found by Cheng and Wei.
Construction of compact embedded $λ$-hypersurfaces via isoparametric hypersurfaces
本文构造了任意非负λ下拓扑为S^1×M的闭嵌入λ-超曲面，其中M为等参超曲面，推广了先前结果并补全了Riedler构造的缺失情形。
In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere $S^{n+1}(1)$ and construct many compact nontrivial embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m>0$ in $S^{n+1}(1)$, for $1\leq m\leq n-1$. In particular, if the $4^{\text{th}}$ mean curvature $H_4$ takes value between $\dfrac{1}{(\tan \fracπ{k})^4}$ and $\dfrac{k^4-4}{n(n-4)}$ for any integer $k\geq3$, then there exists an $n$-dimensional ($n\geq 5$) compact nontrivial embedded hypersurface with constant $H_4$ in $S^{n+1}(1)$.
We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth $(n+1)$--dimensional Riemannian manifolds, a theorem proved first by Pitts for $2\leq n\leq 5$ and extended later by Schoen and Simon to any $n$.
On the First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere
针对单位球面中闭嵌入极小超曲面，改进了其Laplace-Beltrami算子第一非零特征值的下界，结果略优于Duncan-Sire-Spruck的界。
For a compact connected Lie group $G$ acting as isometries on a compact orientable Riemannian manifold $M^{n+1},$ and cohomogeneity not equal to 0 or 2, we prove the existence of a nontrivial embedded $G$-invariant minimal hypersurface, that is smooth outside a set of Hausdorff dimension at most $n-7.
Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren-Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min-max theories.
Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. We prove that a compact embedded hypersurface without boundary in $\R^{n+1}$ with $H^F_r={constant}$ is the Wulff shape, up to translations and homotheties. In case $r=1$, our result is the anisotropic version of Alexandrov Theorem, which gives an affirmative answer to an open problem of F. Morgan.
The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture
本文证明单位球面中嵌入极小超曲面的第一非零特征值等于其维数，从而肯定解决 Yau 猜想，并由此推导出多个刚性定理。
We prove that on a compact Riemannian manifold of dimension $3$ or higher, with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by the first author and N. Wickramasekera in 2020), with prescribing function taken to be a non-zero constant $λ$, produces an embedded hypersurface of constant mean curvature $λ$ ($λ$-CMC). More precisely, we prove that the interface arising from said min-max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of the aforementioned work by the fir…
We extend to the eigenvalues of the hypersurface Spin$^c$ Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Futhermore, we prove a correspondence between the existence of a Spin$^c$ Killing spinor on homogeneous 3-dimensional manifolds $\mathbb E^*(κ, τ)$ with 4-dimensional isometry group and isometric immersions of $\mathbb E^*(κ, τ)$ into the complex space form $\mathbb M^4(c)$ of constant holomorphic sectional curvature $4c$, for some $c\in \mathbb R^*$. As applications, we show the non-existence of totally umbilic surfaces in $\mathbb E^*(κ, τ)$ a…
Embedded cylindrical and doughnut-shaped $λ$-hypersurfaces
本文构造了圆柱状非凸完备嵌入 $\lambda$-超曲面以及两个不等距的紧致甜甜圈形嵌入 $\lambda$-超曲面，说明 $\lambda$-超曲面不满足自收缩子的平面区域猜想相关结论。
In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m$ in a unit sphere $S^{n+1}(1)$ and prove that if the $m^{\text{th}}$ mean curvature $H_m$ takes value between $\dfrac{1}{(\tan \fracπ{k})^m}$ and $\frac{k^2-2}{n}(\frac{k^2+m-2}{n-m})^{\frac{m-2}{2}}$ for $1\leq m\leq n-1$ and any integer $k\geq 2$, then there exists at least one $n$-dimensional compact nontrivial embedded hypersurface with constant $H_m>0$ in $S^{n+1}(1)$. When $m=1$, our results reduce to the results of Perdomo \cite{[P]}; when $m=2$ and $m=4$, our results reduce to the re…
Recently Brendle-Huisken introduced a fully nonlinear flow $G$. Their aim was to extend the surgery algorithm of Huisken-Sinestrari, into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface $M_0$ in $\mathbb{R}^{n+1}$ undergoing this $G$-flow. In order to do this we make some adjustments to Brendle and Huiskens gradient estimate, after we have done this we can go on to argue as in Huisken-Sinestrari's paper Mean curvature flow with surgeries of two-convex hyperusrfaces, in order to classify two-convex s…
Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part I
本文构造并考察单位球面中的嵌入CMC超曲面族，为新实例提供依据，并针对$H=0$情形验证Yau关于Clifford超曲面面积最小的猜想。
A congruence theorem for compact embedded hypersurfaces in $\mathbb{S}^{n+1}_+$
该研究借助空间形式的Reilly型公式，针对球面空间中可平均凸等距嵌入到上半球面的紧子流形，建立了余维数约化定理与同余定理，推进了微分几何中子流形刚性问题的研究。
We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold of dimension at most seven, can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems follows our local picture.
A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds
该研究在正里奇曲率流形中，通过曲率界、法向注入半径和超曲面几何特性，推导出嵌入超曲面第一个非零特征值的下界估计，扩展了Choi和Wang的经典结果到非极小情况。
In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves shows that for any compact Riemannian manifolds with nonnegative R…
An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, and it is a generalization of surface area. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wulff shape. We prove that, unlike the isotropic case, there exists an anisotropic energy density function such that there exist closed embedded equilibrium surfaces with genus zero in the three-dimensional Euclidean space each of which is not (any homothety and translation of) the …
Delzant type theorem for torus-equivariantly embedded toric hypersurfaces
本文研究辛环面流形中余维一复子环面轨道闭包的非奇异性条件，以多面体语言刻画，并推广了Delzant定理。
Let m>1 and n>1 be any pair of integers. In this paper we prove that if H is between the numbers \cot(\fracπ{m}) and b_{m,n}=\frac{(m^2-2)\sqrt{n-1}}{n\sqrt{m^2-1}}, then, there exists a non isoparametric, compact embedded hypersurface in S^{n+1} with constant mean curvature H that admits the group O(n)x Z_m in their group of isometries, here O(n) is the set of n x n orthogonal matrices and Z_m are the integers mod m. When m=2 and H is close to the boundary value 0, the hypersurfaces look like two very close n-dimensional spheres with two catenoid necks attached, similar to constructions made …
Hypersurfaces embedded in conformal manifolds appear frequently as boundary data in boundary-value problems in cosmology and string theory. Viewed as the non-null conformal infinity of a spacetime, we consider hypersurfaces embedded in a Riemannian (or Lorentzian) conformal manifold. We construct a finite and minimal family of hypersurface tensors -- the curvatures intrinsic to the hypersurface and the so-called ``conformal fundamental forms'' -- that can be used to construct natural conformal invariants of the hypersurface embedding up to a fixed order in hypersurface-orthogonal derivatives o…
In this note, we generalize a characterization of the Clifford torus due to Ros. Let $f:M\rightarrow S^{n+1}$ be an embedded closed minimal hypersurface. Assume there are $(n+2)$ great hyperspheres of $S^{n+1}$ perpendicular to each other, such that $M$ is symmetric with respect to them. Let $S$ denote the square of the length of the second fundamental form of $f$ and let $\bar S=\frac{1}{Vol(M)}\int_{M} Sd M$ be the average of $S$. Then $\bar S\geq n$ with equality holding if and only if $f$ is the Clifford torus $C_{m,n-m}$. It can be rewritten as a Simons' type theorem: If $0\leq \int_M (n-…
In this paper, firstly, we show the existence of a compact embedded constant mean curvature (CMC) hypersurface $Σ_1$ in $\mathbb{S}^{2n}$ of the type $S^{n-1} \times S^{n-1} \times S^{1}$. Moreover, the hypersurface $Σ_1$ exhibits $O(n)\times O(n)$ symmetry. Secondly, we show that there exists a compact embedded CMC-hypersurface $Σ_2 \subset \mathbb{S}^{3n-1}$ of the type $S^{n-1} \times S^{n-1} \times S^{n-1} \times S^{1}$. These results generalize the results of Carlotto and Schulz.
Cauchy Problem for the Dirac operator on spatially non-compact spacetimes
本文证明了在具有完备类空Cauchy超曲面的全局双曲流形上Dirac算子Cauchy问题的适定性，为未来针对非紧致Cauchy超曲面证明Fredholm性结果奠定基础。
Let $M^{n+1}$ be a closed manifold of dimension $3\leq n+1\leq 7$. We show that for a $C^\infty$-generic metric $g$ on $M$, to any connected, closed, embedded, $2$-sided, stable, minimal hypersurface $S\subset (M,g)$ corresponds a sequence of closed, embedded, minimal hypersurfaces $\{Σ_k\}$ scarring along $S$, in the sense that the area and Morse index of $Σ_k$ both diverge to infinity and, when properly renormalized, $Σ_k$ converges to $S$ as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian $3$-manifods.
In this paper we will prove that for every integer n>1, there exists a real number H_0<-1 such that every H\in (-\infty,H_0) can be realized as the mean curvature of a embedding of H^{n-1}\times S^1 in the (n+1)-dimensional spaces H^{n+1}. For $n=2$ we explicitly compute the value H_0. For a general value n, we provide function ξ_n defined on (-\infty,-1), which is easy to compute numerically, such that, if ξ_n(H)>-2π, then, H can be realized as the mean curvature of a embedding of H^{n-1}\times S^1 in the (n+1)-dimensional spaces H^{n+1}.
We prove that every quaternionic-contact structure can be embedded in a quaternionic manifold and define a second fundamental form for a such embedding.
Given a compact Riemannian manifold with boundary, we prove that the limit of a sequence of embedded, almost properly embedded free boundary minimal hypersurfaces, with uniform area and Morse index upper bound, always inherits a non-trivial Jacobi field. To approach this, we prove a one-sided Harnack inequality for minimal graphs on balls with many holes.
We study the geometry of families of hypersurfaces in Eguchi-Hanson space that arise as complex line bundles over curves in $S^2$ and are three-dimensional, non-compact Riemannian manifolds, which are foliated in Hopf tori for closed curves. They are negatively curved, asymptotically flat spaces, and we compute the complete three-dimensional curvature tensor as well as the second fundamental form, giving also some results concerning their geodesic flow. We show the non-existence of $Ł^p$-harmonic functions on these hypersurfaces for every $p \geq 1$ and arbitrary curves, and determine the infi…
Index of minimal hypersurfaces in real projective spaces
本文研究实射影空间中极小超曲面的 Morse 指标，证明嵌入不稳定单侧极小超曲面的指标下界为 n+2 且由三次等参极小超曲面达到，并借助 Lawson 曲面证明三维情形存在任意奇数指标的闭嵌入双侧极小曲面。
Min-max theory for free boundary G-invariant minimal hypersurfaces
本文将自由边界极小极大理论推广到等变情形，证明余齐性≥3的紧李群等距作用下的紧黎曼流形中存在非平凡G-不变带自由边界极小超曲面，且在里奇曲率非负、边界严格凸时有无穷多此类超曲面。
In 3-dimensional Euclidean space, Scherk second surfaces are singly periodic embedded minimal surfaces with four planar ends. In this paper, we obtain a natural generalization of these minimal surfaces in any higher dimensional Euclidean space ${\R}^{n+1}$, for $n \geq 3$. More precisely, we show that there exist $(n-1)$-periodic embedded minimal hypersurfaces with four hyperplanar ends. The moduli space of these hypersurfaces forms a 1-dimensional fibration over the moduli space of flat tori in ${\R}^{n-1}$. A partial description of the boundary of this moduli space is also given.
Analyticity and tautness of smooth Dupin hypersurfaces
本文证明满足Dupin条件的光滑浸入欧氏超曲面局部实解析且半代数，并由此推出紧致嵌入超曲面具有紧致性。
An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere
该研究针对球面中闭嵌入极小超曲面的Laplace-Beltrami算子第一非零特征值，给出了带显式维数常数、依赖第二基本形式长度上界的改进下界，是无额外假设下对Choi & Wang下界的首个可显式计算的提升。
We prove the existence of a one parameter family of minimal embedded hypersurfaces in $R^{n+1}$, for $n \geq 3$, which generalize the well known 2 dimensional "Riemann minimal surfaces". The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar ends. By opposition with the 2-dimensional case, they are not foliated by spheres.
In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. Suppose the union of non-principal orbits $M\setminus M^{reg}$ is a smooth embedded submanifold of $M$ without boundary and ${\rm dim}(M\setminus M^{reg})\leq n-2 $. Then for any $c\in\mathbb{R}$, we show the existence of a nontrivial, smooth, closed, $G$-equivariant almost embedded $G$-invariant hypersurface $Σ^n$ of constant mean curvature $c$.
In this work, we study various properties of embedded hypersurfaces in $1+1+2$ decomposed spacetimes with a preferred spatial direction, denoted $e^μ$, which are orthogonal to the fluid flow velocity of the spacetime and admit a proper conformal transformation. To ensure a non-vanishing positivity scalar curvature of the induced metric, we impose that the scalar curvature of the conformal metric is non-negative and that the associated conformal factor satisfies $\hatφ^2+2\hat{\hatφ}>0$, where $\hat{\ast}$ denotes derivative along $e^μ$. Firstly, it is demonstrated that such hypersurface is eit…
We prove a non-collapsing property for curvature flows of embedded hypersurfaces in the sphere and in hyperbolic space.
We study stable immersed capillary hypersurfaces in a domain $\mathcal B$ which is either a half-space or a slab in the Euclidean space $\Bbb R^{n+1}.$ We prove that such a hypersurface $Σ$ is rotationally symmetric in the following cases: (1) $n=2$, $\mathcal B$ is a slab and $Σ$ has genus zero, (2) $n\geq 2$, $\mathcal B$ is a slab, the angle of contact is $π/2$ and each component of $\partialΣ$ is embedded, (3) $n\geq 2,$ $\mathcal B$ is a half-space, the angle of contact is $<π/2$ and each component of $\partialΣ$ is embedded. Moreover, in case (2), if not a right circular cylinder, the hy…
In this paper, we construct an immersed, non-embedded $S^{n}$ $λ$-hypersurface in Euclidean spaces $\mathbb{R}^{n+1}$.
We construct new examples of embedded, complete minimal hypersurfaces in quaternionc hyperbolic space and also some minimal foliations. We introduce fans an construct analytic deformations of bisectors.
Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal curvatures
Topological Half-Space Theorems for Minimal Hypersurfaces in $\mathbb{R}^4$
研究 \(\mathbb{R}^4\) 中极小超曲面的拓扑伯恩斯坦定理，证明曲率有界、与 \(\mathbb{R}^3\) 微分同胚且在平板中的完备恰当嵌入极小超曲面必为超平面，立方体积增长假设下半空间中的也成立。
We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant $δ(n)>0$ depending only on $n$ such that on any closed embedded, non-totally geodesic, minimal hypersurface $M^n$ in $\mathbb{S}^{n+1}$, $$\int_{M}S \geq δ(n){\rm Vol}(M^n),$$ where $S$ is the squared length of the second fundamental form of $M^n$. The Perdomo Conjecture asserts that $δ(n)=n$ which is still open in general. As byproducts, we also obtain some integral inequalities and Simons-type pinching results on closed embedded (or immersed) minimal hypersurfaces, with the first positive eigenvalue $λ_1(…
We prove a new upper bound for the smallest eigenvalues of the Dirac operator on a compact hypersurface of the hyperbolic space.
In the proof of the BFK-gluing formula for zeta-determinants of Laplacians there appears a real polynomial whose constant term is an important ingredient in the gluing formula. This polynomial is determined by geometric data on an arbitrarily small collar neighborhood of a cutting hypersurface. In this paper we express the coefficients of this polynomial in terms of the scalar and principal curvatures of the cutting hypersurface embedded in the manifold when this hypersurface is 2-dimensional. Similarly, we express some coefficients of the heat trace asymptotics of the Dirichlet-to-Neumann ope…
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