Marcos Dajczer, Ruy Tojeiro
We solve Blaschke's problem for hypersurfaces of dimension $n\geq 3$. Namely, we determine all pairs of Euclidean hypersurfaces $f,\tilde{f}\colon M^n\to\R^{n+1}$ that induce conformal metrics on $M^n$ and envelope a common sphere congruence in $\R^{n+1}$.
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Let $x: \mathbf{M} \rightarrow {\mathbb Q}^{n+1}_1$ be a regular space-like hypersurface in the conformal space ${\mathbb Q}^{n+1}_1$. We classify all those hypersurfaces with parallel Blaschke tensor in the conformal space up to the conformal equivalence.
In this paper, we give a complete conformal classification of the regular space-like hypersurfaces in the de Sitter Space $\mathbb{S}^{m+1}_{1}$ with parallel para-Blaschke tensors.
In [2] we have classified the Blaschke quasi-umbilical submanifolds in the conformal space ${\mathbb Q}^n_s$. In this paper we shall classify the Blaschke para-umbilical hypersurfaces in the conformal space ${\mathbb Q}^n_s$. That may be also considered as the extension of the classification of the conformal isotropic submanifolds in the conformal space ${\mathbb Q}^n_s$.
Let $M^n$ be an $n$-dimensional umbilic-free hypersurface in the $(n+1)$-dimensional Lorentzian space form $M^{n+1}_1(c)$. Three basic invariants of $M^n$ under the conformal transformation group of $M^{n+1}_1(c)$ are a $1$-form $C$, called conformal $1$-form, a symmetric $(0,2)$ tensor $B$, called conformal second fundamental form, and a symmetric $(0,2)$ tensor $A$, called Blaschke tensor. The so-called para-Blaschke tensor $D^λ=A+λB$, the linear combination of $A$ and $B$, is still a symmetric $(0,2)$ tensor. A spacelike hypersurface is called a para-Blaschke isoparametric spacelike hypersu…
In this paper, we use two conformal non-homogeneous coordinate systems, modeled on the de Sitter space ${\mathbb S}^{m+1}_1$, to cover the conformal space ${\mathbb Q}^{m+1}_1$, so that the conformal geometry of regular space-like hypersurfaces in $\mathbb{Q}^{m+1}_1$ is treated as that of hypersurfaces in ${\mathbb S}^{m+1}_1$. As a result, we give a complete classification of the regular space-like hypersurfaces (represented in the de Sitter space ${\mathbb S}^{m+1}_1$) with parallel Blaschke tensors.
The volume distance from a point p to a convex hypersurface M of the (N+1)-dimensional space is defined as the minimum (N+1)-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane H(p) we obtain, after normalization, a symmetric bi-linear form Q. In this paper, we prove that Q converges to the affine Blaschke metric when we approximate the hypersurface along a curve whose points are centroids of parallel sections. We also show that the rate of this convergence is gi…
Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this tensor for locally strongly convex hypersurfaces M, n>2, with two distinct affine principal curvatures or with three distinct affine principal curvatures assuming that at least one affine principal curvature has multiplicity 1.
The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n > 2. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n=3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant, and therefore are either spherical caps or pieces of tubes (canal surfaces).
We prove that among all constant width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant width body obtained by rotation of the Reuleaux triangle about an axis of symmetry.
We consider the generalization of classical Blaschke's Problem to higher codimension case, characterizing Darboux pair of isothermic surfaces and dual S-Willmore surfaces as the only non-trivial surface pairs that envelop a 2-sphere congruence and conformally correspond to each other. When the sphere congruence is the mean curvature spheres of one envelop surface, it must be a cmc-1 surface in hyperbolic 3-space, or a S-Willmore surface. A study of conformally immersed surface pairs is indicated with discussion on the geometric meaning of new invariants.
We show that isothermic surfaces and S-Willmore surfaces are also the solutions to the corresponding Blaschke's problem for both spacelike and timelike surfaces in pseudo-Riemannian space forms. For timelike surfaces both Willmore and isothermic, we obtain a description by minimal surfaces similar to the classical results of Thomsen.
In this paper we study the linearizability problem for 3-webs on a 2-dimensional manifold. With an explicit computation based on the theory developed in the paper "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp. 2643-2654), we examine a 3-web whose linearizability was claimed in the same paper. We show that, contrary to the statement of the papers "On the Blaschke conjecture for 3-webs" (arXiv: math.DG/0411460) and "On linearization of planar three-webs and Blaschke's conjecture", (C.R.Acad. Sci. Paris, Ser. I. vol. 341. num 3 (2005)), this particular web is linearizable. …
In this paper, our purpose is to study rigidity theorems for $λ$-hypersurfaces in Euclidean space under Gauss map. As a Bernstein type problem for $λ$-hypersurfaces, we prove that an entirely graphic $λ$-hypersurface in Euclidean space is a hyperplane.
The problem of determining the {\it Bonnet hypersurfaces in} $R^{n+1}$, for $n>1$, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called {\it Bonnet associate} of the initial hypersurface. The orthogonal net which is called \hbox{\it $A$-net} is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in $R^{n+1}$ …
We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled real hypersurfaces in nonflat complex space forms are minimal, which provides their classification due to a known result of Lohnherr and Reckziegel.
We prove the existence of minimal hypersurfaces for the Dirichlet that extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian ambient manifolds
We give new characterisations of sets of positive reach and show that a closed hypersurface has positive reach if and only if it is of class $C^{1,1}$. These results are then used to prove new alternating Steiner formulæ for hypersurfaces of positive reach. Furthermore, it will turn out that every hypersurface that satisfies an alternating Steiner formula has positive reach. Finally, we provide a new solution to a problem by Hadwiger on convex sets and prove long time existence for the gradient flow of mean breadth.
It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.
We consider the problem $F=f(ν)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.
In this paper we deal with the following problem: Find all Riemannian metrics on a manifold that can be realized isometrically as immersed hypersurfaces in the Euclidean space. We study this problem for a wide class of metrics on hypersurfaces arising from Codazzi tensors.
We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.
We consider the problem $F=f(ν)$ for strictly convex, closed hypersurfaces in hyperbolic space and solve it for curvature functions $F$ the inverses of which are of class $(K^*)$.
Given a C2-domain with compact boundary in an arbitrary complete Riemannian manifold, we search for smallness conditions on the boundary data for which the Dirichlet problem for the minimal hypersurface equation is solvable. We obtain an extension to Riemannian manifolds of an existence result of G. H. Williams ( J. Reine Angew. Math. 354:123-140, 1984).
Characterizations of umbilical hypersurfaces by partially overdetermined problems in space forms
研究空间形式中 Serrin 型部分超定问题对脐超曲面的刚性刻画，将齐次 Robin 边界条件推广为非齐次，从而把接触角从正交推广到任意 $\theta\in(0,\pi)$。
A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map $f$ of the open unit disk there is even an indestructible Blaschke product $B$ such that the critical sets of $f$ and $B$ coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighte…
We study constant mean curvature spacelike hypersurfaces in generalized Robertson-Walker spacetimes which are spatially parabolic covered (i.e. its fiber F is a (non- compact) complete Riemannian manifold whose universal covering is parabolic) and satisfy the null convergence condition. In particular, we provide several rigidity results under appro- priate mathematical and physical assumptions. We pay special attention to the case where the GRW spacetime is Einstein. As an application, some Calabi-Bernstein type results are given.
A compact approach to the positivity of Brown-York's mass and its relation with the Min-Oo conjecture, Yau's Problem \#100 and rigidity of hypersurfaces
本文提出一种紧致方法探讨Brown-York质量的正性，并揭示其与Min-Oo猜想、Yau问题#100及超曲面刚性的内在联系。
Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-α$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then, given a compact topological $G-$shaped hypersurface $Γ$ in the asymptotic boundary of $M,$ and $|H|<\sqrtα$, we prove the existence of a complete properly embedded hypersurface whose mean curvature is equal to $H$ and whose asymptotic boundary is $Γ$. We are able, this way, to extend a previous theorem of B.Guan and J.Spruck on the hyperbolic space to any rank 1 symm…
Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.
In this note we motivate the definition and use of Lie algebroids by revisiting the problem of reconstructing a hypersurface in Euclidean space from infinitesimal data.
We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions.
In this paper, the pinching problems of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$ are studied. By making use of the Sobolev inequality, we prove a global pinching theorem of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$.
This is Part 1 of two papers where we develop the basic potential theory of elliptic operators on posssibly singular almost minimzers using their hyperbolic unfoldings. We can establish surprisingly robust boundary Harnack inequalities along the singular set. We apply them to derive a Martin theory and solve classical boundary value problems and we also consider eigenvalue problems and variational problems on such singular spaces.
Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces
该研究提出半空间中各向异性毛细超曲面的新Hsiung-Minkowski积分公式，证明相关Alexandrov型定理与刚性结果，得到各向异性Orlicz-Christoffel-Minkowski问题解的唯一性，为p≥1的L_p-Minkowski问题解的唯一性提供新证明。
In a round ball $B\subset\mathbb{R}^{n+1}$ endowed with an $O(n+1)$-invariant metric we consider a radial function that weights volume and area. We prove that a compact two-sided hypersurface in $B$ which is stable capillary in weighted sense and symmetric about some line containing the center of $B$ is homeomorphic to a closed $n$-dimensional disk. When combined with Hsiang symmetrization and other stability results this allows to deduce that the interior boundary of any isoperimetric region in $B$ for the Gaussian weight is a closed $n$-disk of revolution. For $n=2$ we also show that a compa…
We show that for a very general and natural class of curvature functions (for example the curvature quotients $(σ_n/σ_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(κ) = σ\in (1,\infty)$ with a prescribed compact future asymptotic boundary $Γ$ at infinity has at least one smooth solution (if l = 1 or l = 2 there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly con…
Hypersurfaces of constant scalar curvature in hyperbolic space with prescribed asymptotic boundary at infinity
本文推广双曲空间经典渐近普拉托问题，通过构造常数量曲率图极限并引入新技术建立二阶先验估计，证明所有可能曲率值下存在具指定无穷远渐近边界的光滑完备常数量曲率超曲面。
We study the isometric spacelike embedding problem in scaled de Sitter space, and obtain Weyl-type estimates and the corresponding closedness in the space of embeddings.
We develope in great computational details the classical Cartan equivalence problem for Levi-nondegenerate C^6-smooth real hypersurfaces M^3 in C^2, performing all calculations effectively in terms of a (local) graphing function φ. In particular, we present explicitly the unique (complex) essential invariant J of the problem. Its expansion in terms of the 3-variables function φincorporates millions of differential monomials, while, when φis assumed to depend only on 2 variables (rigid case), J writes out in two lines (7 monomials).
Homogeneous hypersurfaces of the four-dimensional Thurston geometry ${\rm Sol_0^4}$
本文对四维Thurston几何 ${\ m Sol_0^4}$ 中具有常主曲率的超曲面进行分类，并以此给出了该几何中齐次超曲面的完整分类，解决了Erjavec和Inoguchi提出的相关问题。
Willmore-type variational problem for foliated hypersurfaces
本文研究带叶状结构超曲面的Willmore型变分问题，建立Reilly型泛函，计算第一、第二变分并证明部分共形不变性，推导临界超曲面的Euler-Lagrange方程，并构造旋转临界超曲面且证明其局部极小性。
In the present paper, we handle the problem of finding a hypersurface family from a given asymptotic curve in R^4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.
Do Carmo's problem for CMC hypersurfaces in $\mathbb{R}^6$
本文证明 $\mathbb{R}^6$ 中有限指标完备非紧常均曲率超曲面必为极小，肯定回答 do Carmo 问题，并推广至 $\mathbb{R}^4$、$\mathbb{R}^5$。
Prescribed $L_{p}$ curvature problem for convex capillary hypersurface
本文研究了欧几里得半空间中凸毛细超曲面的 prescribed $L_p$ 曲率问题，通过转化问题为 Hessian 商类型方程，证明了解的存在性和唯一性，并应用该方法解决了毛细 $L_p$ Christoffel-Minkowski 问题及相关特征值问题。
In this paper, we study the Dirichlet problem for $p$-convex hypersurfaces with prescribed curvature. We prove that there exists a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition. An interior curvature estimate is also obtained.
Exponential bounds for the spherical Blaschke-Lebesgue problem
研究高维单位球面上常宽凸体的最小体积，通过构造和Schramm论证的推广，给出相对有效半径的非平凡上下界。
A Lipschitz hypersurface is a hypersurface which locally is the graph of a Lipschitz function. A Lipschitz (or C^1) hypersurface is said to be Levi-flat if it is locally foliated by complex manifolds of complex dimension (n-1). We shall prove that there exist no Lipschitz Levi-flat hypersurfaces in CP^n with n >= 3. Our new estimates on the d-bar-Cauchy problems are different from the earlier Siu's integral kernal method.
Formula for the force field of Newtonian dynamical systems admitting the normal shift of hypersurfaces in Riemannian manifolds is considered. Problem of globalization for geometric structures associated with this formula is studied.
Convex capillary hypersurfaces of prescribed curvature problem
本文将闭凸超曲面的规定$k$-阶Weingarten曲率问题扩展到毛细情形，通过将其转化为球冠上带Robin边界条件的Hessian商方程可解性问题，在自然充分条件下证明了严格凸毛细超曲面的存在性，扩展了毛细Minkowski问题的相关研究。
Problem of global integration of geometric structures arising in the theory of dynamical systems admitting the normal shift is considered. In the case when such integration is possible the problem of globalization for shift maps is studied.
In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a large family of uniformly elliptic EDP's, so solving the Calabi-Bernstein problem in a wide family of spacetimes.
For a congruence of straight lines defined by a hypersurface in $R^{n+1}, n \geq 1,$ and a field of reflected directions created by a point source we define the notion of intensity in a tangent direction and introduce elementary symmetric functions $S_m, m=1, 2,...,n,$ of {\it principal intensities}. The problem of existence and uniqueness of a closed hypersurface with prescribed $S_n$ is the "reflector problem" extensively studied in recent years. In this paper we formulate and give sufficient conditions for solvability of an analogous problem in which the mean intensity $S_1$ is a given func…
We study the equivalence problem under projective transformation for CR-hypersurfaces of complex projective space. A complete set of projective differential invariants for analytic hypersurfaces is given. The self-dual strongly C-linearly convex hypersurfaces are characterized.
Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on $§^n$ for conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory inte…
Alexandrov-type theorem for singular capillary CMC hypersurfaces in the half-space
本文研究了半空间中相对等周型问题临界点的分类，通过引入新型移位距离函数，在特定正则性假设下证明了奇异毛细管CMC超曲面的Alexandrov型定理。
The $L_p$ dual Minkowski problem for capillary hypersurfaces
本文研究了$L_p$对偶Minkowski问题，通过转化为Monge-Ampère方程并证明在特定条件下存在唯一解，解决毛细超曲面的曲率测度问题。
In this article spacelike hypersurfaces immersed in twisted product spacetimes $I\times_f F$ with complete fiber are studied. Several conditions ensuring global hyperbolicity are presented, as well as a relation that needs to hold on each spacelike hypersurface in $I\times_f F$ for it to be a simple warped product. When the fiber is assumed to be closed (compact and without boundary) and the ambient spacetime has a suitable expanding behaviour, non-existence results for constant mean curvature hypersurfaces are obtained. Under the same hypothesis, a characterization of compact maximal hypersur…
In this paper, we study the exterior problem for the maximal surface equation. We obtain the precise asymptotic behavior of the exterior solution at infinity. And we prove that the exterior Dirichlet problem is uniquely solvable given admissible boundary data and prescribed asymptotic behavior at infinity.
In this paper, we establish the curvature estimates for $p$-convex hypersurfaces in $\mathbb{R}^{n+1}$ of prescribed curvature with $p\geq \frac{n}{2}$. The existence of a star-shaped hypersurface of prescribed curvature is obtained. We also prove a type of interior $C^2$ estimates for solutions to the Dirichlet problem of the corresponding equation.
We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordström manifolds.
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