The Willmore functional and the containment problem in R4

Let Σ be a convex hypersurface in the Euclidean space R ⁴ with mean curvature H. We obtain a geometric lower bound for the Willmore functional ∫_Σ H ² dσ. This bound is an invariant involving the area of Σ, the volume and Minkowski quermassintegrals of the convex body that Σ bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R ⁴.     

The relative isoperimetric inequality outside convex domains in R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We prove that the area of a hypersurface Σ which traps a given volume outside a convex domain C in Euclidean space R ⁿ is bigger than or equal to the area of a hemisphere which traps the same volume on one side of a hyperplane. Further, when C has smooth boundary ∂C, we show that equality holds if and only if Σ is a hemisphere which meets ∂C orthogonally.     

The Willmore functional and the containment problem in R~4

Let∑be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional∫∑H2dσ. This bound is an invariant involving the area of∑, the volume and Minkowski quermassintegrals of the convex body that∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.     

The Blow-up Locus of Heat Flows for Harmonic Maps

Abstract Let M and N be two compact Riemannian manifolds. Let u _k (x, t) be a sequence of strong stationary weak heat flows from M×R ⁺ to N with bounded energies. Assume that u _k→u weakly in H ^{1, 2}(M×R ⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^t is an H ^m−2-rectifiable set for almost all t∈R ⁺. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σ^t is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪_t<0Σ^t× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion. This work is supported by NSF grant     

Reduced bodies in normed planes

We say that a convex body R of a d-dimensional real normed linear space M ^d is reduced, if Δ(P) < Δ(R) for every convex body P ⊂ R different from R. The symbol Δ(C) stands here for the thickness (in the sense of the norm) of a convex body C ⊂ M ^d . We establish a number of properties of reduced bodies in M ². They are consequences of our basic Theorem which describes the situation when the width (in the sense of the norm) of a reduced body R ⊂ M ² is larger than Δ(R) for all directions strictly between two fixed directions and equals Δ(R) for these two directions.     

Singular sets of convex bodies and surfaces with generalized curvatures

In this paper we consider generalized surfaces with curvature measures and we study the properties of those k-dimensional subsets Σ ^k of such surfaces where the curvatures have positive density with respect to k-dimensional Hausdorff measure. Special attention is given to boundaries of convex bodies inR ³. We introduce a class of convex sets whose curvatures live only on integer dimension sets. For such convex sets we consider integral functionals depending on the curvature and the area ofK and on the curvature andH ^k of Σ ^k .     

Marginally Trapped Surfaces in L<Superscript>4</Superscript> and an Extended Weierstrass–Bryant Representation

Let E be the total space of a locally trivial torus bundle over a surface Σ_g of genus g > 1. Using Seiberg–Witten theory and spectral sequences, we prove that E carries a symplectic structure if and only if the homology class of the fiber [T²] is nonzero in H₂(E, R). Mathematics Subject Classifications (2000): 53C42, 53A10.     

Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l.     

The Bernstein problem for affine maximal hypersurfaces

In this paper, we prove the validity of the Chern conjecture in affine geometry [18], namely that an affine maximal graph of a smooth, locally uniformly convex function on two dimensional Euclidean space, R ², must be a paraboloid. More generally, we shall consider the n-dimensional case, R ⁿ , showing that the corresponding result holds in higher dimensions provided that a uniform, “strict convexity” condition holds. We also extend the notion of “affine maximal” to non-smooth convex graphs and produce a counterexample showing that the Bernstein result does not hold in this generality for dimension n≥10. Oblatum 16-IV-1999 & 4-XI-1999?Published online: 21 February 2000     

Topology of the hyperspace of convex bodies of constant width

The hyperspace of all convex bodies of constant width in Euclidean spaceR ⁿ ,n≥2, is proved to be homeomorphic to a contractibleQ-manifold (Q denotes the Hilbert cube). The proof makes use of an explicitly constructed retraction of the entire hyperspace of convex bodies on the hyperspace of convex bodies of constant width. Translated fromMaternaticheskie Zametki, Vol. 62, No. 6, pp. 813–819, December, 1997 Translated by V. N. Dubrovsky     

Classification of the isomorphic types of martingale-H 1 spaces

LetF _n be an increasing sequence of finite fields on a probability space (Ω,F _n,P) whereF denotes the σ-algebra generated by ∪F _n. ThenF _n is isomorphic to one of the following spaces:H ¹(δ), ΣH _n ¹ ,l ^l.     

Fourier Inversion of Distributions Supported by a Hypersurface

Let μ _Σ be the natural measure on R ^N (N≥3) supported by a compact oriented analytic hypersurface Σ, ψ a smooth function on R ^N and P(D) a differential operator in N variables of order m. We determine a sufficient condition on the number λ such that the Fourier integral of the distribution P(D)ψ μ _Σ be summable by Cesàro means of order λ to zero in a point outside the hypersurface. This condition depends on m and on the position of the point with respect to the caustic of the hypersurface.     

Critical Points of Wang�CYau Quasi-Local Energy

In this paper, we prove the following theorem regarding the Wang–Yau quasi-local energy of a spacelike two-surface in a spacetime: Let Σ be a boundary component of some compact, time-symmetric, spacelike hypersurface Ω in a time-oriented spacetime N satisfying the dominant energy condition. Suppose the induced metric on Σ has positive Gaussian curvature and all boundary components of Ω have positive mean curvature. Suppose H ≤ H ₀ where H is the mean curvature of Σ in Ω and H ₀ is the mean curvature of Σ when isometrically embedded in \mathbb R³{\mathbb R^3} . If Ω is not isometric to a domain in \mathbb R³{\mathbb R^3}, then

Lipschitz selections of set-valued mappings and Helly’s theorem

We prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family. Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most 2^m+1 points, the restriction F|_M′ of F to M′ has a selection f_M′ (i. e., f_M′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒ_M′(x) − ƒ_M′(y)‖_X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖_X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper bound of the number of points in M′, 2^m+1, is sharp. If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition for a function defined on a closed subset of R ² to be the restriction of a function from the Sobolev space W _∞ ² (R ²).A similar result is proved for the space of functions on R ² satisfying the Zygmund condition.     

Approximation of analytic by Borel sets and definable countable chain conditions

LetI be a σ-ideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ ₁ ¹ countable chain condition if there is aΣ ₁ ¹ equivalence relation with uncountably many equivalence classes none of which is inI. Assuming definable determinacy, we show that if the family of Borel sets fromI is definable in the codes of Borel sets, then eachΣ ₁ ¹ set is equal to a Borel set modulo a set fromI iffI fulfils theΣ ₁ ¹ countable chain condition. Further we characterize the σ-idealsI generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property forΣ ₁ ¹ sets mentioned above. It turns out that they are exactly of the formMGR(F)={A : ∀F ∈ F A ∩F is meager inF} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager sets is the unique σ-ideal onR, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition. Research partially supported by NSF grant DMS-9317509.     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

Concentration sets of the Landau–Lifshitz system and quasi-mean curvature flows

The aim of this work is to analyze the concentration set of the stationary weak solutions to the Landau-Lifshitz system of the ferromagnetic spin chain from R ^m into the unit sphere S ² of R ³ . Suppose that u _k → u weakly in W ^1,2(R ^m × R ₊, S ²) and that Σ ^t is the concentration set for fixed t. In the present paper we first prove that Σ ^t is a -rectifiable set for almost all t ∈ R ₊. And then we verify that Σ ^t moves by the quasi-mean curvature under some assumptions, which is a new codimension 2 curvature flow. Finally we analyze the behavior of the solution at the singular point and get the blow up formulas. The main barrier to Landau–Lifshtiz system is that there is no energy monotonicity inequality. After the seminal work to on the study of the concentration set of minimizing energy harmonic maps by Leon Simon, there are several papers dealing with the stationary harmonic maps and its heat flows, and so on. This investigation is inspired by the study on the heat flow of harmonic maps and it largely depends on our result of the partial regularity.     

A Polynomial-Time Algorithm to Approximate the Mixed Volume within a Simply Exponential Factor

Let K=(K ₁,…,K _n ) be an n-tuple of convex compact subsets in the Euclidean space R ⁿ , and let V(⋅) be the Euclidean volume in R ⁿ . The Minkowski polynomial V _K is defined as V _K (λ ₁,…,λ _n )=V(λ ₁ K ₁+⋅⋅⋅+λ _n K _n ) and the mixed volume V(K ₁,…,K _n ) as

On the Semicontinuity of Curvature Integrals

Let H_j(K, ·) be the j – th elementary symmetric function of the principal curvatures of a convex body K in Euclidean d – space. We show that the functionals ∫_bd f(H_j(K, x)) dℋ︁^d—1(x) depend upper semicontinuously on K, if f : [0, ∞) is concave, lim_t→0f(t) = 0, and lim_t→∞f(t)/t = 0. An analogous statement holds for integrals of elementary symmetric functions of the principal radii of curvature.     

Average bending of convex pleated planes in hyperbolic three-space

If P is a pleated plane in 3-dimensional hyperbolic space H ³ and α a geodesic in its intrinsic metric we define B(P,α), the average bending of P in the direction α. We show that if P is a convex pleated plane embedded in H ³ then B(P,α)≤K for some universal K. Furthermore if P_Γ is a boundary component of the convex hull of a quasi-Fuchsian group Γ then B(P_Γ,α)=B(Γ) almost everywhere, where B(Γ) is a constant times the length of the bending lamination β_Γ of the pleated surface X _Γ=P_Γ/Γ. We use these to prove a number of results about quasi-Fuchsian groups including a universal bound on the Lipschitz constant for the map to infinity and a bound on the length of β_Γ by a constant times the Euler characteristic of X _Γ. Oblatum 10-X-1996 & 23-V-1997