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Register a new userOn constant curvature submanifolds of space forms
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We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $fcolon M^n_ctoQ^{n+p}_{tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature $tilde c$, and free of weak-umbilic points if $c>tilde{c}$. We show that the substantial codimension of $f$ is $p=n-1$ if, as shown by Cartan and Moore, the first normal bundle possesses the lowest possible rank $n-1$. These submanifolds are of a class that has been extensively studied due to their many properties. For instance, they are holonomic and admit B{a}cklund and Ribaucour transformations.
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We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $mgeq 3$ of a product $Q_{k_1}^{n_1}times Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2 eq 0$ to the classification of $m$-dimensional umbilical submanifolds of codimension two of $Sf^ntimes R$ and $Hy^ntimes R$. The case of $Sf^ntimes R$ was carried out in cite{mt}. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.
The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,ldots,k_{n-1})$ satisfying suitable conditions. In this paper we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator $f$ is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is quantified in terms of the oscillation of the curvature function $f$. Our approach provides a unified picture of quantitative studies of the method of moving planes in space forms.
We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the $L$-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the $L$-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequence, we derive a Bianchi-cube theorem, which allows to produce, from $k$ initial scalar $L$-transforms of a given submanifold of constant curvature, a whole $k$-dimensional cube all of whose remaining $2^k-(k+1)$ vertices are submanifolds with the same constant sectional curvature given by explicit algebraic formulae. We also obtain further reductions, as well as corresponding decomposition and Bianchi-cube theorems, for the classes of $n$-dimensional flat Lagrangian submanifolds of $mathbb{C}^n$ and $n$-dimensional Lagrangian submanifolds with constant curvature $c$ of the complex projective space $mathbb Cmathbb P^n(4c)$ or the complex hyperbolic space $mathbb Cmathbb H^n(4c)$ of complex dimension $n$ and constant holomorphic curvature~4c.
We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetricinequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $n=2$ case of a conjecture of Gir~ao-Pinheiro.
In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical $n$-dimensional minimal submanifolds with index of relative nullity $n-2$, fully described by Dajczer and Florit cite{DF2} in terms of a certain class of elliptic surfaces. Opposed to this, we prove that nonminimal $n$-dimensional submanifolds in space forms of any codimension are locally cylinders provided that they carry a totally geodesic distribution of rank $n-2geq2,$ which is contained in the relative nullity distribution, such that the length of the mean curvature vector field is constant along each leaf. The case of dimension $n=3$ turns out to be special. We show that there exist elliptic three-dimensional submanifolds in spheres satisfying the above properties. In fact, we provide a parametrization of three-dimensional submanifolds as unit tangent bundles of minimal surfaces in the Euclidean space whose first curvature ellipse is nowhere a circle and its second one is everywhere a circle. Moreover, we provide several applications to submanifolds whose mean curvature vector field has constant length, a much weaker condition than being parallel.
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