Laguerre Geometry of Hypersurfaces in Rn\R^n

Laguerre geometry of surfaces in $\R^3$ is given in the book of Blaschke [1], and have been studied by E.Musso and L.Nicolodi [5], [6], [7], B. Palmer [8] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in $\R^n$. For any umbilical free hypersurface $x: M\to\R^n$ with non-zero principal curvatures we define a Laguerre invariant metric $g$ on $M$ and a Laguerre invariant self-adjoint operator ${\mathbb S}: TM\to TM$, and show that $\{g,{\mathbb S}\}$ is a complete Laguerre invariant system for hypersurfaces in $\R^n$ with $n\ge 4$. We calculate the Euler-Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space $\R^n$, the Lorentzian space $\R^n_1$ and the degenerate space $\R^n_0$ we define three Laguerre space forms $U\R^n$, $U\R^n_1$ and $U\R^n_0$ and define the Laguerre embedding $U\R^n_1\to U\R^n$ and $U\R^n_0\to U\R^n$, analogue to the Moebius geometry where we have Moebius space forms $S^n$, \H^n and $\R^n$ (spaces of constant curvature) and conformal embedding \H^n\to S^n and $\R^n\to S^n$ (cf. [4], [10]). Using these Laguerre embedding we can unify the Laguerre geometry of hypersurfaces in $\R^n$, $\R^n_1$ and $\R^n_0$. As an example we show that minimal surfaces in $\R^3_1$ or $\R_0^3$ are Laguerre minimal in $\R^3$.Comment: 24 page

Deformation of Hypersurfaces Preserving the Moebius Metric and a Reduction Theorem

A hypersurface without umbilics in the n+1 dimensional Euclidean space is known to be determined by the Moebius metric and the Moebius second fundamental form up to a Moebius transformation when n>2. In this paper we consider Moebius rigidity for hypersurfaces and deformations of a hypersurface preserving the Moebius metric in the high dimensional case n>3. When the highest multiplicity of principal curvatures is less than n-2, the hypersurface is Moebius rigid. Deformable hypersurfaces and the possible deformations are also classified completely. In addition, we establish a Reduction Theorem characterizing the classical construction of cylinders, cones, and rotational hypersurfaces, which helps to find all the non-trivial deformable examples in our classification with wider application in the future.Comment: 51 pages. A mistake in the proof to Theorem 9.2 has been fixed. Accepted by Adv. in Mat

M\"obius and Laguerre geometry of Dupin Hypersurfaces

In this paper we show that a Dupin hypersurface with constant M\"{o}bius curvatures is M\"{o}bius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hypersurface. These results solve the major issues related to the conjectures of Cecil et al on the classification of Dupin hypersurfaces.Comment: 45 pages. arXiv admin note: text overlap with arXiv:math/0512090 by other author

Complete self-shrinkers with bounded the second fundamental form in Rn+1\mathbb{R}^{n+1}

Let $X:M^n\to \mathbb{R}^{n+1}$ be a complete properly immersed self-shrinker. In this paper, we prove that if the squared norm of the second fundamental form $S$ satisfies $1\leq S< C$ for some constant $C$, then $S=1$. Further we classify the $n$-dimensional complete proper self-shrinkers with constant squared norm of the second fundamental form in $\mathbb{R}^{n+1}$, which solve the conjecture proposed by Q.M. Cheng and G. Wei when the self-shrinker is proper.Comment: 17 pages, 0 figure, All comments are welcom