A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom. 56:1–66, 2000; Math. Ann. 333: 131–155, 2005) and Morgan–Tian ( [math.DG], 2008). A version of Perelman’s collapsing theorem states: “Let {M³_i}\{M^{3}_{i}\} be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and $\mathrm{diam}(M^{3}_{i})\ge c_{0}>0$\mathrm{diam}(M^{3}_{i})\ge c_{0}>0 . Suppose that all unit metric balls in M³_iM^{3}_{i} have very small volume, at most v _i →0 as i→∞, and suppose that either M³_iM^{3}_{i} is closed or has possibly convex incompressible toral boundary. Then M³_iM^{3}_{i} must be a graph manifold for sufficiently large i”. This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman’s critical point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds.     

Some generalizations of Backʼs Theorem

The problem of the existence of jointly continuous utility functions is studied. A continuous representation theorem of Back [1] gives the existence of a continuous map from the space of total preorders topologized by closed convergence (Fell topology) to the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact-open topology. The commodity space is locally compact and second countable. Our results generalize Back?s Theorem to non-metrizable commodity spaces with a family of not necessarily total preorders. Precisely, we consider regular commodity spaces having a weaker locally compact second countable topology.     

On Perelman’s dilaton

By means of a Kaluza–Klein type argument we show that the Perelman’s F{mathcal{F}} -functional is the Einstein–Hilbert action in a space with extra “phantom” dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his famous paper (Perelman in The entropy formula for the Ricci flow and its geometric applications, 2002). As a consequence the Ricci flow (modified by a diffeomorphism and a time-dependent factor) is the evolution of the “real” part of the metric under a constrained gradient flow of the Einstein–Hilbert gravitational action in higher dimension.     

A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \(\mathbf {RCA}_0\) to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements.     

Combinatorial Generalizations of Jung’s Theorem

We consider combinatorial generalizations of Jung’s theorem on covering a set by a ball. We prove the “fractional” and “colorful” versions of the theorem.     

A Generalisation of Tverberg’s Theorem

We will prove the following generalisation of Tverberg’s Theorem: given a set S⊂ℝ ^d of (r+1)(k−1)(d+1)+1 points, there is a partition of S in k sets A ₁,A ₂,…,A _k such that for any C⊂S of at most r points, the convex hulls of A ₁\C,A ₂\C,…,A _k \C are intersecting. This was conjectured first by Natalia García-Colín (Ph.D. thesis, University College of London, 2007).     

Symmetric Versions of Laman’s Theorem

Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are “fixed” by various symmetry operations of the framework.     

On Analogues of Heilbronn’s Theorem

Mathematical Notes - Continued fractions with rational partial quotients arise in a natural way in the course of applying any $$k$$ -ary gcd algorithm to the ratio of natural numbers $$a$$ , $$b$$...     

A generalization of Menger’s Theorem

This paper generalizes one of the celebrated results in Graph Theory due to Karl. A. Menger (1927), which plays a crucial role in many areas of flow and network theory. This paper also introduces and characterizes strength reducing sets of nodes and arcs in weighted graphs.     

Perelman’s lambda-functional and the stability of Ricci-flat metrics

In this article, we introduce a new method (based on Perelman’s λ-functional) to study the stability of compact Ricci-flat metrics. Under the assumption that all infinitesimal Ricci-flat deformations are integrable we prove: (a) a Ricci-flat metric is a local maximizer of λ in a C ^2,α -sense if and only if its Lichnerowicz Laplacian is nonpositive, (b) λ satisfies a ?ojasiewicz-Simon gradient inequality, (c) the Ricci flow does not move excessively in gauge directions. As consequences, we obtain a rigidity result, a new proof of Sesum’s dynamical stability theorem, and a dynamical instability theorem.     

On second variation of Perelman’s Ricci shrinker entropy

In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman??s ??-entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of (Cao in Adv Lect Math 11:1?C38, 2010). Moreover, we obtain a necessary condition for linearly stable shrinkers in terms of the least eigenvalue and its multiplicity of certain Lichnerowicz type operator associated to the second variation.     

On a Generalization of Voronin’s Theorem

Mathematical Notes - Voronin’s theorem states that the Riemann zeta-function ζ(s) is universal in the sense that all analytic functions that are defined and have no zeros on the right...     

A Complex Analogue of Toda’s Theorem

Toda (SIAM J. Comput. 20(5):865–877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P ^#P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of Blum–Shub–Smale real machines (Blum et al. in Bull. Am. Math. Soc. 21(1):1–46, 1989) was proved in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010). Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) is topological in nature; the properties of the topological join are used in a fundamental way. However, the constructions used in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) were semi-algebraic—they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) to the complex projective case. A key role is played by the complex join of quasi-projective complex varieties. As a consequence, we obtain a complex analogue of Toda’s theorem. The results of this paper, combined with those in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010), illustrate the central role of the Poincaré polynomial in algorithmic algebraic geometry, as well as in computational complexity theory over the complex and real numbers: the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.     

A General Version of Price’s Theorem

Journal of Theoretical Probability - Assume that $$X_{Sigma } in mathbb {R}^{n}$$ is a centered random vector following a multivariate normal distribution with positive definite covariance...     

Irreducibility and extensions of Ostrowski’s Theorem

In this paper an extension of Ostrowski’s Theorem for complex square irreducible matrices is presented. Also extensions of similar statements for square complex matrices are analyzed and completed. Most of the statements in this work cover also the case of reducible matrices.     

A simple proof of Veech’s Theorem

We present a simple proof of Veech’s Theorem.     

Smale’s Fundamental Theorem of Algebra Reconsidered

In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s upper bound estimate was infinite average cost. Ours is polynomial in the Bézout number and the dimension of the input. Hence it is polynomial for any range of dimensions where the Bézout number is polynomial in the input size. In particular it is not just for the case that Smale considered but for a range of dimensions as considered by Bürgisser–Cucker, where the max of the degrees is greater than or equal to n ^1+? for some fixed ?. It is possible that Smale’s algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem.     

A Bicategorical Version of Masuoka’s Theorem

We give a bicategorical version of the main result of Masuoka (Tsukuba J Math 13:353–362, 1989) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings R í SR \subseteq S, the relative Picard group Pic(S/R) is isomorphic to the Amitsur 1–cohomology group H ¹(S/R,U) with coefficients in the units functor U.     

A generalization of Chaplygin’s Reducibility Theorem

In this paper we study Chaplygin’s Reducibility Theorem and extend its applicability to nonholonomic systems with symmetry described by the Hamilton-Poincaré-d’Alembert equations in arbitrary degrees of freedom. As special cases we extract the extension of the Theorem to nonholonomic Chaplygin systems with nonabelian symmetry groups as well as Euler-Poincaré-Suslov systems in arbitrary degrees of freedom. In the latter case, we also extend the Hamiltonization Theorem to nonholonomic systems which do not possess an invariant measure. Lastly, we extend previous work on conditionally variational systems using the results above. We illustrate the results through various examples of well-known nonholonomic systems.