A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications

Integral section formulae for totally geodesic submanifolds (planes) intersecting a compact submanifold in a space form are available from appropriate representations of the motion invariant density (measure) of these planes. Here we present a new decomposition of the invariant density of planes in space forms. We apply the new decomposition to rewrite Santaló's sectioning formula and thereby to obtain new mean values for lines meeting a convex body. In particular we extend to space forms a recently published stereological formula valid for isotropic plane sections through a fixed point of a convex body in R³.     

Comparison Theorems on Convex Hypersurfaces in Hadamard Manifolds

In a Hadamard manifold with sectional curvaturebounded from below by –k ₂ ², we give sharp upper estimates for the difference circumradius minus inradiusof a compact k ₂-convex domain, and we getalso estimates for the quotient (Total d-mean curvature)/Area of a convex domain.     

A geometric inequality on mixed volumes

We develop some geometric inequality for a kind of generalized convex set. The integral of (n – 2)-th mean curvature of the generalized convex set, the mixed volume of the convex hull of the set, and a reference convex set are involved in the inequality.Partially supported by grants from Kosef and BSRI-95-1419.     

Absolute curvature measures,II

The absolute curvature measures for sets of positive reach in R ^d introduced in [7] satisfy the following kinematic relations: Their integrated values on the intersections with (or on the tangential projections onto) uniformly moved p-planes are constant multiples of the corresponding absolute curvature measures of the primary set. In the special case of convex bodies the first result is the so-called Crofton formula. An analogue for signed curvature measures is well known in the differential geometry of smooth manifolds, but the motion of absolute curvatures used there does not lead to this property. For the special case of smooth compact hypermanifolds our absolute curvature measures agree with those introduced by Santaló [4] with other methods.In the appendix, the section formula is applied to motion invariant random sets.     

Width of convex bodies in spaces of constant curvature

We consider the measure of points, the measure of lines and the measure of planes intersecting a given convex body K in a space form. We obtain some integral formulas involving the width of K and the curvature of its boundary ∂K. Also we study the special case of constant width. Moreover we obtain a generalisation of the Heintze–Karcher inequality to space forms. Work partially supported by grant number ACI2003-44 Joint action Catalonia–Baden-Württemberg and by FEDER/MEC grant number MTM2006-04353. The third author was also supported by the program Ramón y Cajal, MEC.     

On the volume functional of compact manifolds with boundary with constant scalar curvature

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.     

The Gauss-Bonnet theorem and Crofton-type formulas in complex space forms

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in any complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives the Gauss curvature integral in terms of the Euler characteristic, and some hermitian intrinsic volumes. The other one, which is shorter, involves the measure of complex hyperplanes meeting the domain. As a tool, we obtain variation formulas in integral geometry of complex space forms.     

Hypersurfaces whose mean curvature function is of finite type

We define finite mean type hypersurfaces to be hypersurfaces with mean curvature function of finite Chen-type. Then, we prove that hyperplanes are the only polynomial translation hypersurfaces of finite mean type in a Euclidean spaceE ⁿ⁺¹. And we show that the only non-conic hyperquadrics of finite mean type in Euclidean spaces are the hyperspheres and the cylinders on spheres. Finally, we state that, among all hypercylinders in a Euclidean spaceE ⁿ⁺¹, the only ones of finite mean type are those on finite mean type planar curves.     

Reverse Bonnesen style inequalities in a surface \mathbb{X}_\varepsilon ^2 of constant curvature

We investigate the isoperimetric deficit upper bound, that is, the reverse Bonnesen style inequality for the convex domain in a surface $\mathbb{X}_\varepsilon ^2$ of constant curvature ? via the containment measure of a convex domain to contain another convex domain in integral geometry. We obtain some reverse Bonnesen style inequalities that extend the known Bottema’s result in the Euclidean plane $\mathbb{E}^2$ .     

Convex Bodies in Exceptional Relative Positions

Two convex bodies K and K' in Euclidean space Eⁿ can be saidto be in exceptional relative position if they have a commonboundary point at which the linear hulls of their normal coneshave a non-trivial intersection. It is proved that the set ofrigid motions g for which K and gK' are in exceptional relativeposition is of Haar measure zero. A similar result holds trueif ‘exceptional relative position’ is defined viacommon supporting hyperplanes. Both results were conjecturedby S. Glasauer; they have applications in integral geometry.     

Singularities of Solution Surfaces for Quasilinear First-Order Partial Differential Equations

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.     

Boundary Measures for Geometric Inference

We study the boundary measures of compact subsets of the d-dimensional Euclidean space, which are closely related to Federer’s curvature measures. We show that they can be computed efficiently for point clouds and suggest that these measures can be used for geometric inference. The main contribution of this work is the proof of a quantitative stability theorem for boundary measures using tools of convex analysis and geometric measure theory. As a corollary we obtain a stability result for Federer’s curvature measures of a compact set, showing that they can be reliably estimated from point-cloud approximations.     

Hypersurfaces in Hn and the space of its horospheres

A classical theorem, mainly due to Aleksandrov [Al2] and Pogorelov [P], states that any Riemannian metric on S ² with curvature K > —1 is induced on a unique convex surface in H ³ . A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another metric on immersed surfaces, which we call the horospherical metric.?This result extends in higher dimensions, the metrics obtained are then conformally flat. One can also study equivariant immersions of surfaces or the metrics obtained on the boundaries of hyperbolic 3-manifolds. Some statements which are difficult or only conjectured for the induced metric or the third fundamental form become fairly easy when one considers the horospherical metric, which thus provides a good boundary condition for the construction of hyperbolic metrics on a manifold with boundary.?The results concerning the third fundamental form are obtained using a duality between H ³ and the de Sitter space . In the same way, the results concerning the horospherical metric are proved through a duality between H ⁿ and the space of its horospheres, which is naturally endowed with a fairly rich geometrical structure. Submitted: March 2001, Revised: November 2001.     

The affine complete hypersurfaces of constant Gauss-Kronecker curvature

Given a bounded convex domain Ω with C∞ boundary and a function ψ∈C∞（δΩ）, Li-Simon-Chen can construct an Euclidean complete and W-complete convex hypersurface M with constant affine Gauss-Kronecker curvature, and they guess the M is also affine complete. In this paper, we give a confirmation answer.     

Polytopes that fill ℝn and scissors congruenceand scissors congruence

Suppose thatP is a (not necessarily convex) polytope in ℝ ⁿ that can fill ℝ ⁿ with congruent copies of itself. Then, except for its volume, all its classical Dehn invariants for Euclidean scissors congruence must be zero. In particular, in dimensions up to 4, any suchP is Euclidean scissors congruent to ann-cube. An analogous result holds in all dimensions for translation scissors congruence.     

New affine measures of symmetry for convex bodies

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with “enough symmetries”.To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are “far apart”. In particular, we give an example of a convex body whose centroid and Santaló point are “far apart”.     

Splitting and pinching for convex sets

We consider noncompact, closed and convex sets with nonvoid interior in Euclidean space. It is shown that if such a set has one curvature measure sufficiently close to the boundary measure, then it is congruent to a product of a vector space and a compact convex body. Related stability and characterization theorems for orthogonal disc cylinders are proved. Our arguments are based on the Steiner-Schwarz symmetrization processes and generalized Minkowski integral formulas.     

Monge-Ampère foliations with singularities at the boundary of strongly convex domains

Let be a bounded strongly convex domain with smooth boundary. We consider a Monge-Ampère type equation in D with a simple pole at the boundary. Using the Lempert foliation of D in extremal discs, we construct a solution u whose level sets are boundaries of horospheres. Among other things, we show that the biholomorphisms between strongly convex domains are exactly those maps which preserves our solution.     

A criterion for the unique determination of domains in Euclidean spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains

We say that a domain U ⊂ ℝ ⁿ is uniquely determined by the relative metric (which is the extension by continuity of the intrinsic metric of the domain on its boundary) of its Hausdorff boundary if any domain V ⊂ ℝ ⁿ such that its Hausdorff boundary is isometric in the relative metric to the Hausdorff boundary of U, is isometric to U in the Euclidean metric. In this paper, we obtain the necessary and sufficient conditions for the uniqueness of determination of a domain by the relative metric of its Hausdorff boundary.     

Minimal total absolute curvature for immersions

Immersions or maps of closed manifolds in Euclidean space, of minimal absolute total curvature are called tight in this paper. (They were called convex in [25].) After the definition in Chapter 1, many examples in Chapter 2, and some special topics in Chapter 3, we prove in Chapter 4 that topological tight immersions ofn-spheres are only of the expected type, namely embeddings onto the boundary of a convexn+1-dimensional body. This generalises a theorem of Chern and Lashof in the smooth case. In Chapter 5 we show that many manifolds exist that have no tight smooth immersion in any Euclidean space.This research was partially supported by National Science Foundation grant GP-7952X1.