On the Patterns of Principal Curvature Lines around a Curve of Umbilic Points

Surfaces with one constant principal curvature in three-dimensional space forms

We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal plane. In some cases, a kind of trichotomy phenomenon is observed: the curves of the foliations may be circles, hyperbolas or horocycles, depending of whether the constant principal curvature is respectively larger, smaller of equal to one (not necessarily in this order). We descr…

Umbilic Singularities and Lines of Curvature on Ellipsoids of ${\mathbb R}^4$

The topological structure of the lines of principal curvature, the umbilic and partially umbilic singularities of all tridimensional ellipsoids of ${\mathbb R}^4$ is described.

Partially umbilic singularities of hypersurfaces of $\mathbb R^4$

This paper establishes the geometric structure of the lines of principal curvature of a hypersurface immersed in ${\mathbb R}^4$ in a neighborhood of the set $\mathcal{S}$ of its principal curvature singularities, consisting of the points at which atF least two principal curvatures are equal. Under generic conditions defined by appropriate transversality hypotheses it is proved that $\mathcal{S}$ is the union of regular smooth curves $\mathcal{S}_{12}$ and $\mathcal{S}_{23}$, consisting of partially umbilic points, where only two principal curvatures coincide. This curve is partitioned into re…

脐斜率与三次Weingarten曲面

Umbilic slopes and cubic Weingarten surfaces

本文研究脐斜率并分类满足κ₂=cκ₁³的紧致浸入曲面，证明其每个连通分量均为旋转椭球，方法涉及齐次Hessian恒等式、唯一延拓及四阶增长估计。

空间形式中位于全脐点曲面上的曲线的刻画

A Characterization of Curves that Lie on a Totally Umbilical Surface of a Space Form

该研究刻画了空间形式$\boldsymbol{\rm H}^3$和$\boldsymbol{\rm S}^3$中位于全脐点曲面上的正则曲线，给出其曲率与挠率的充要条件，还得到常挠率曲线的曲率值并展示数值图像。

The umbilic set of Willmore surfaces

It is well known that the umbilic points of minimal surfaces in spaces of constant sectional curvature consist only of isolated points unless the surface is totally umbilic on some connected component, as for example the Hopf form is holomorphic. In this note, we prove that on Willmore surfaces in codimension one the umbilic set is locally a one dimensional real-analytic manifold without boundary or an isolated point.

解析曲面的退化脐点

Degenerate Umbilic Points of Analytic Surfaces

本文通过更简单的几何论证证明解析曲面上脐点指数不能大于1，并基于退化性建立指数下界，为Carathéodory猜想提供新见解。

平面伪测地线与全脐子流形

Planar pseudo-geodesics and totally umbilic submanifolds

本文研究 Riemannian 流形间的全脐等距浸入，通过引入平面伪测地线这一自然外蕴推广，给出了具有平行单位平均曲率向量的全脐等距浸入的新刻画，并证明了余维为一时浸入全脐当且仅当所有测地线的外蕴形状为平面，推广了三维欧氏空间曲面的经典结果。

On the umbilic set of immersed surfaces in three-dimensional space forms

We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a $3$-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.

Totally quasi-umbilic timelike surfaces in $\mathbb{R}^{1,2}$

For a regular surface in Euclidean space $\mathbb{R}^3$, umbilic points are precisely the points where the Gauss and mean curvatures $K$ and $H$ satisfy $H^2=K$; moreover, it is well-known that the only totally umbilic surfaces in $\mathbb{R}^3$ are planes and spheres. But for timelike surfaces in Minkowski space $\mathbb{R}^{1,2}$, it is possible to have $H^2=K$ at a non-umbilic point; we call such points {\em quasi-umbilic}, and we give a complete classification of totally quasi-umbilic timelike surfaces in $\mathbb{R}^{1,2}$.

Affine curvature lines of surfaces in 3-space

In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines in the neighborhood of points with double eigenvalues (but not umbilics) of the affine shape operator and parabolic points

Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with umbilic boundary

Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that the boundary has constant mean curvature. In this paper we address to the stability of this problem with respect of perturbation of mean curvature of the boundary and scalar curvature of the manifold. In particular we prove that the Yamabe boundary problem is stable under perturbation of the mean curvature and the scalar curvature from below, while it is not stable if one of the two curvatures is perturbed from above.

全非线性曲率流的局部脐点、凸性和柱面估计

Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

本文将Huisken--Stampacchia迭代局部化方法推广至全非线性曲率流，建立了曲率一次齐次函数演化超曲面的渐近尖锐曲率夹挤估计局部化，并扩展至弯曲环境空间。

The index of isolated umbilics on surfaces of non-positive curvature

It is shown that if a $C^2$ surface $M\subset\mathbb R^3$ has negative curvature on the complement of a point $q\in M$, then the $\mathbb Z/2$-valued Poincaré-Hopf index at $q$ of either distribution of principal directions on $M-\{q\}$ is non-positive. Conversely, any non-positive half-integer arises in this fashion. The proof of the index estimate is based on geometric-topological arguments, an index theorem for symmetric tensors on Riemannian surfaces, and some aspects of the classical Poincaré-Bendixson theory.

Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with non umbilic boundary

We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a la…

Codimension two Umbilic points on Surfaces Immersed in R^3

In this paper is studied the behavior of lines of curvature near umbilic points that appear generically on surfaces depending on two parameters.

Spacelike capillary surfaces in the Lorentz-Minkowski space

For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz-Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilic point, which was developed by Choe \cite{Choe}. Using the concept of the rotation index at the interior and boundary umbilic points and applying the Poincaré-Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than $4$ vertices in a domain of $\Bbb L^3$ bounded by (spacelike or timelike) totally umbilic surfaces is part of a (spacelike) pla…

Rigidity of submanifolds with parallel mean curvature in space froms

Let $M$ be an $n(\geq3)$-dimensional oriented compact submanifold with parallel mean curvature in the simply connected space form $F^{n+p}(c)$ with $c+H^2>0$, where $H$ is the mean curvature of $M$. We prove that if the Ricci curvature of $M$ satisfies $Ric_{M}\geq(n-2)(c+H^2),$ then $M$ is either a totally umbilic sphere, the Clifford hypersurface $S^{m}\big(\frac{1}{\sqrt{2(c+H^2)}}\big)\times S^{m}\big(\frac{1}{\sqrt{2(c+H^2)}}\big)$ in $S^{n+1}(\frac{1}{\sqrt{c+H^2}})$ with $n=2m$, or $\mathbb{C}P^{2}(4/3(c+H^2))$ in $S^7(\frac{1}{\sqrt{c+H^2}})$. In particular, if $Ric_{M}>(n-2)(c+H^2),$ …

Classification of Hypersurfaces with Two Distinct Principal Curvatures and Closed Moebius Form in $\mathbb{S}^{m+1}$

Let $x$ be an $m$-dimensional umbilic-free hypersurface in an $(m+1)$-dimensional unit sphere $\mathbb{S}^{m+1}(m\geq3)$. One of important questions is to classify hypersurfaces with two distinct principal curvatures. In this paper, we classify and explicitly express the hypersurfaces with two distinct principal curvatures and closed Moebius form, and then we characterize and classify conformally flat hypersurfaces of dimension larger than 3.

Curvature lines of a transversal eqüiaffine vector field along a surface in $3$-space

In this paper we discuss the behavior of the curvature lines of a transversal eqüiaffine vector field along a surface in $3$-space at isolated umbilical points.

关于3D半黎曼流形中null螺旋的一些讨论

On the invariant trajectories of lightlike paths near central potential singularities in semi-Riemannian manifolds of index 2

本文探讨了三维半黎曼流形中完全脐子流形上null螺旋的几何性质，通过分析null弗莱内框架和退化度量条件，推导出表征null螺旋的方程和不变量，为相关理论提供新贡献。

Characterizing a surface by invariants

Canonical principal parameters are introduced for surfaces in $\mathbb R^3$ without umbilical points. It is proved that in these parameters the surface is determined (up to position in space) by a pair of invariants satisfying a partial differential equation equivalent to the Gauss equation. As such a pair of invariants we may use the principal curvatures or the Gauss and the mean curvature.

Geometry of surfaces in $\mathbb R^5$ through projections and normal sections

We study the geometry of surfaces in $\mathbb R^5$ by relating it to the geometry of regular and singular surfaces in $\mathbb R^4$ obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which are not second order geometry for surfaces in $\mathbb R^5$ but are in $\mathbb R^4$. We also relate the umbilic curvatures of each type of surface and their contact with spheres. We then consider the surfaces as normal sections of 3-manifolds in $\mathbb R^6$ and again relate asymptotic directions and contact with spheres by defining an appropriate umbilic …

类时CMC曲面上的Hopf微分与曲率线流

Hopf differentials and curvature line flows on time-like CMC surfaces

该研究探讨洛伦兹3空间形式中类时CMC曲面上Hopf微分与曲率线流的关联，证明非退化Hopf微分下脐点处曲率线流的指标由其阶数模4的余数确定，丰富了类时曲面的几何性质研究。

Remarks on the boundary curve of a constant mean curvature topological disc

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a condition for the surface to be totally umbilic in terms of the normal curvature.

Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows

In the 1950's Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper we investigate conditions under which evolving a smooth rotationally symmetric sphere by a linear combination of its radii of curvature yields a Hopf sphere. When the coefficients of the flow have certain integer values, the fate of an initial sphere is entirely determined by the local geometry of its isolated umbilic points. A surprising variety of behaviours is uncovered: convergence to round spheres …

Harmonic Mean Curvature Lines on Surfaces Immersed in R3

Consider oriented surfaces immersed in $\mathbb R^3.$ Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature $\mathcal K$, given by the product of the principal curvatures $k_1, k_2$ is positive. The leaves of the foliations are the lines of harmonic mean curvature, also called characteristic or diagonal lines, along which the normal curvature of the immersion is given by ${\mathcal K}/{\mathcal H}$, where $ {\mathcal H}=({k_1}+k_2)/2$ is the arithmetic mean curvature. That is, ${\mathcal K}/{\mathca…

Lines of axial curvature at critical points on surfaces mapped into R4

In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter families of mappings of surfaces into R4. As the parameter crosses a critical bifurcation value, at which the mapping has a critical point, it is described how the axial umbilic points, which are the singularities of the axial curvature configurations at regular points, move alo…

Weingarten曲面的性质与变换

Properties and Transformations of Weingarten Surfaces

本文研究E3中旋转对称Weingarten曲面，分析脐点斜率限制、SL2(R)几何变换及Lagrangian变分表述，并证明半二次型曲面上作用的传递性与稳定性。

Normal curvatures of asymptotically constant graphs and Caratheodory's conjecture

We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the normal curvatures of these graphs, which establish some weaker forms of the conjecture. In particular, we show that there are uncountably many principal lines in the graph of f whose projection into R^2 are parallel to any given direction.

Triharmonic CMC hypersurfaces in space forms with at most 3 distinct principal curvatures

A $k$-harmonic map is a critical point of the $k$-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^{n} (n\ge 3)$ is a CMC proper triharmonic hypersurface with at most three distinct principal curvatures in a space form $\mathbb{R}^{n+1}(c)$, then $M$ has constant scalar curvature. This supports the generalized Chen's conjecture when $c\le 0$. When $c=1$, we give an optimal upper bound of the mean curvature $H$ for a non-totally umbilical proper CMC $k$-harmonic hypersurface with constant scalar curvature in a sphere. As an application, we …

Isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces

In this paper, we study the relationship between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces. We give some examples of isoparametric hypersurfaces with (non)constant principal curvatures on Randers manifolds with nonconstant flag curvatures. Furthermore, we construct an example of a conformally flat Randers manifold which admits a family of nonisoparametric hyperplanes with constant principal curvatures.

Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.

Geometric Mean Curvature Lines on Surfaces Immersed in R3

Here are studied pairs of transversal foliations with singularities, defined on the Elliptic region (where the Gaussian curvature $\mathcal K$ is positive) of an oriented surface immersed in $\mathbb R^3$. The leaves of the foliations are the lines of geometric mean curvature, along which the normal curvature is given by $\sqrt {\mathcal K}$, which is the geometric mean curvature of the principal curvatures $ k_1, k_2$ of the immersion. The singularities of the foliations are the umbilic points and parabolic curves}, where $ k_1 = k_2$ and ${\mathcal K} = 0$, respectively. Here are determined …

Real hypersurfaces with constant principal curvatures in complex hyperbolic spaces

We classify all real hypersurfaces with three distinct constant principal curvatures in complex hyperbolic spaces of dimension greater than two.

The constancy of principal curvatures of curvature-adapted submanifolds in symmetric spaces

In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under certain condition, and derive the constancy of the principal curvatures of such submanifolds. As its result, we can derive that such submanifolds are isoparametric.

On the curvature of G-manifolds with finitely many non-principal orbits

We investigate the curvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature and non-negative sectional curvature, and discuss some families of examples to which these existence results apply.

Behavior of principal curvatures of frontals near non-front singular points and their application

We investigate behavior of principal curvatures and principal vectors near a non-degenerate singular point of the first kind of frontals. As an application, we extend the notion of Ribaucour transformations to frontals with singular points.

复双曲二次型中具有常主曲率的Hopf超曲面

On Hopf hypersurfaces of the complex hyperbolic quadric with constant principal curvatures

本文对复双曲二次型中具有常主曲率的Hopf超曲面进行分类，并确定了三或四个不同常主曲率时的曲率值及重数。

Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes

We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian flat surfaces with parallel mean curvature or, equivalently, by principal orbits of a cohomogeneity two polar action.

A nonisoparametric hypersurface with constant principal curvatures

In this note we construct an explicit example of a (compact) conformally flat Riemannian manifold which admits a totally geodesic foliation of codimension one with no isoparametric leaves. This answers negatively the question: is every hypersurface with constant principal curvatures isoparametric?

Biharmonic hypersurfaces in space forms with three distinct principal curvatures

In this paper, we have studied biharmonic hypersurfaces in space form $\bar{M}^{n+1}(c)$ with constant sectional curvature $c$. We have obtained that biharmonic hypersurfaces $M^{n}$ with at most three distinct principal curvatures in $\bar{M}^{n+1}(c)$ has constant mean curvature. We also obtain the full classification of biharmonic hypersurfaces with at most three distinct principal curvatures in arbitrary dimension space form $\bar{M}^{n+1}(c)$.

Dupin hypersurfaces with four principal curvatures, II

If $M$ is an isoparametric hypersurface in a sphere $S^n$ with four distrinct principal curvatures, then the principal curvatures $κ_1,...,κ_4$ can be ordered so that their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and the cross-ratio $r$ of the principal curvatures (the Lie curvature) equals -1. In this paper, we prove that if $M$ is an irreducible connected proper Dupin hypersurface in $\R^n$ (or $S^n$) with four distinct principal curvatures with multiplicities $m_1=m_2 \geq 1$ and $m_3=m_4=1$, and constant Lie curvature $r=-1$, then $M$ is equivalent by Lie sphere transformation to a…

主曲率曲面上的几何

Geometry on principal curvature surfaces

研究与惠特尼伞相关的主曲率曲面几何，证明其与法平面交线和焦圆锥性质，通过研究多种曲率沿例外集的情况，确定曲率函数零点数量。

Manifolds with almost nonnegative curvature operator and principal bundles

We study manifolds with almost nonnegative curvature operator (ANCO) and provide first examples of closed simply connected ANCO mannifolds that do not admit nonnegative curvature operator.

Focal surfaces of fronts associated to unbounded principal curvatures

We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near non-degenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of singularities and geometrical properties of initial fronts. Moreover, we investigate behavior of the Gaussian curvature of the focal surface.

Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to open parts of tubes around the ruled minimal submanifolds with totally real normal bundle introduced by Berndt and Bruck. In particular, they are open parts of homogenous ones.

Principal curvatures and parallel surfaces of wave fronts

We give criteria for which a principal curvature becomes a bounded $C^\infty$-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.

H-hypersurfaces with at most 3 distinct principal curvatures in the Euclidean spaces

In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures. We also give explicit examples.

复二次曲面上具有常主曲率的Hopf超曲面

On Hopf hypersurfaces of the complex quadric with constant principal curvatures

本文分类了复二次曲面上具有至多五个互异常主曲率以及低维情形下具有常主曲率的Hopf超曲面，并指出它们均为齐次例子的开部分。

Hypersurfaces in $CP^2$ and $CH^2$ with two distinct principal curvatures

It is known that hypersurfaces in $CP^n$ or $CH^n$ for which the number $g$ of distinct principal curvatures satisfied $g \le 2$ must belong to a standard list of Hopf hypersurfaces with constant principal curvatures, provided that $n \ge 3$. In this paper, we construct a 2-parameter family of non-Hopf hypersurfaces in $CP^2$ and $CH^2$ with $g=2$ and show that every non-Hopf hypersurface with $g=2$ is locally of this form.

Biharmonic hypersurfaces with three distinct principal curvatures in spheres

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…

在 $\mathbb{H}^m\times \mathbb{H}^n$ 中具有常主曲率的超曲面的分类

Classification of hypersurfaces with constant principal curvatures in $\mathbb{H}^m\times \mathbb{H}^n$

本文对双曲空间乘积中具有常主曲率的超曲面进行分类，证明了常乘积角函数情形下主曲率个数至多为3，并给出完整分类及等参超曲面的推论。

Volume preserving flow by powers of symmetric polynomials in the principal curvatures

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm. In the case of the volume preserving scalar curvature flow, we can prove that the hypersurfaces converge smoothly and exponentially fast to a round sphere.