Rotational symmetry and properties of the ancient solutions of Ricci flow on surfaces

We give a simple proof for the rotational symmetry of ancient solutions of Ricci flow on surfaces. As a consequence we obtain a simple proof of some results of Daskalopoulos, Hamilton and Sesum on the a priori estimates for the ancient solutions of Ricci flow on surfaces. We also give a simple proof for the solution to be a Rosenau solution under some mild conditions on the solutions of Ricci flow on surfaces.     

A note on uniformization of Riemann surfaces by Ricci flow

We clarify that the Ricci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.

     

Remark on a lower diameter bound for compact shrinking Ricci solitons

In this paper, inspired by Fernández-López and García-Río [11], we shall give a new lower diameter bound for compact non-trivial shrinking Ricci solitons depending on the range of the potential function, as well as on the range of the scalar curvature. Moreover, by using a universal lower diameter bound for compact non-trivial shrinking Ricci solitons by Chu and Hu [7] and by Futaki, Li, and Li [13], we shall provide a new sufficient condition for four-dimensional compact non-trivial shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Furthermore, we shall give a new lower diameter bound for compact self–shrinkers of the mean curvature flow depending on the norm of the mean curvature. We shall also prove a new gap theorem for compact self–shrinkers by showing a necessary and sufficient condition to have constant norm of the mean curvature.     

Heat kernel bounds, ancient κ solutions and the Poincaré conjecture

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3-dimensional ancient κ solutions to the Ricci flow. As an application, using the W entropy associated with the heat kernel, we give a different and much shorter proof of Perelman's classification of backward limits of these ancient solutions. The method is partly motivated by Cao (2007) [1] and Sesum (2006) [27]. The current paper or Chow and Lu (2004) [6] combined with Chen and Zhu (2006) [4] and Zhang (2009) [31] lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume.     

关于二维球面上的Ricci流的一个注记

给出Hamilton和Chow关于二维球面上的Ricci流的一个定理的新证明．这个证明结合了Hamilton关于Ricci流的紧性定理和二维ancientp解的分类和Perelman的非塌缩定理．     

On the Equivalence of Minimax Theorems

In this note we show that Ky Fan's minimax theorem and its several generalizations such as König's minimax theorem [6], M. Neumann's minimax theorem [8] and Fuchssteiner-König's minimax theorem [3] are equivalent. We also give a direct proof for Fuchssteiner-König's minimax theorem on the basis of Eidelheit's well-known seperation theorem.     

Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere

We prove a comparison theorem for the isoperimetric profiles of solutions of the normalized Ricci flow on the two-sphere: If the isoperimetric profile of the initial metric is greater than that of some positively curved axisymmetric metric, then the inequality remains true for the isoperimetric profiles of the evolved metrics. We apply this using the Rosenau solution as the model metric to deduce sharp time-dependent curvature bounds for arbitrary solutions of the normalized Ricci flow on the two-sphere. This gives a simple and direct proof of convergence to a constant curvature metric without use of any blowup or compactness arguments, Harnack estimates, or any classification of behaviour near singularities.     

On the Existence of Min-Max Minimal Torus

In this paper, we will study the existence problem of min-max minimal torus. We use classical conformal invariant geometric variational methods. We prove a theorem about the existence of min-max minimal torus in Theorem 5.1. First we prove a strong uniformization result (Proposition 3.1) using the method of Ahlfors and Bers (Ann. Math. 72(2):385–404, 1960). Then we use this proposition to choose good parameterization for our min-max sequences. We prove a compactification result (Lemma 4.1) similar to that of Colding and Minicozzi (Width and finite extinction time of Ricci flow, [math.DG], 2007), and then give bubbling convergence results similar to that of Ding et al. (Invent. math. 165:225–242, 2006). In fact, we get an approximating result similar to the classical deformation lemma (Theorem 1.1).     

Some remarks on Arslan’s 2011 paper

It is shown that the main theorem of Arslan’s paper (Theorem 2, 2011), as stated, is incorrect. Under additional conditions, we present a short proof of the corrected version of the theorem. We also give a proof of a theorem of Rao and Shanbhag (1991) [2], employed by Arslan, without the use of the Kolmogorov Consistency Theorem.     

Forward–backward SDEs with random terminal time and applications to pricing special European-type options for a large investor

In this paper, we first discuss the solvability of coupled forward–backward stochastic differential equations (FBSDEs, for short) with random terminal time. We prove the existence and uniqueness of adapted solution to such FBSDEs under some natural assumptions. The method of proof adopted is to construct a contraction mapping related to the solutions of a sequence of backward SDEs. Our monotonicity-type assumptions are different from those in Hu and Peng (1995) [4], Peng and Shi (2000) [11], and so on. As a corollary of our main result, the solvability of FBSDEs with a finite time horizon is discussed. Finally, the existence and uniqueness theorem of the solution to FBSDEs with a finite time horizon is applied to price special European-type options for a large investor.     

The Kähler Ricci flow on Fano surfaces (I)

Suppose {(M, g(t)), 0 ≤ t < ∞} is a Kähler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kähler Ricci flow on toric Fano surface, whose initial metric has toric symmetry. In particular, such a Kähler Ricci flow must converge to a Kähler Ricci soliton metric. Therefore we give a new Ricci flow proof of the existence of Kähler Ricci soliton metrics on toric Fano surfaces.     

Zum Nachweis arithmetischer Cohen-Macaulay Varietäten

There exist some useful methods for the calculation of Hilbert's function without using a free resolution of polynomial ideals (see for example [4], [10], [11] and the references in these papers). Using Bezout's theorem (in the sense ofW. Gröbner [3], 144.5) these methods are suited for a proof that special homogeneous polynomial ideals are imperfect, but not for the arithmetically Cohen-Macaulay property. It is the theorem of this paper that these gaps can be filled. This theorem therefore provides some proof that an arbitrary homogeneous polynomial ideal is perfect or imperfect. Our methods are demonstrated in three examples, taking the third example from the paper ofG. A. Reisner [7], p. 35 and, using our methods, we rather easily obtain the result of [7], that the Cohen-Macaulay property depends on the characteristic of the field. In the second example, we give some remarks on the usefulness of the definition for perfeet ideals ofF. S. Macaulay [5] (see also [6]). This also illustrates whyF. S. macaulay could only construct imperfect ideals-except such one obtainable by using ideals of the principal class.

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Unserem Lehrer, Herrn Professor Dr. W. Gröbner, zum 80. Geburtstag in Verehrung gewidmet     

随机积分和微分方程解的存在性准则

在[1]中我们已证明了一个一般的随机不动点定理并给出了某些应用,在本文中我们将给出该结果的进一步应用.首先证明了一随机Darbo不动点定理,然后利用此定理在紧性假设下给出了非线性随机Volterra积分方程和非线性随机微分方程Cauchy问题随机解的存在性准则.我们的定理改进和推广了Lakshmikantham[3,4],Vaugham[2],De Blasi和Myjak[5]等人的结果.     

The computational content of Walras’ existence theorem

In [Y. Tanaka, Undecidability of the Uzawa equivalence theorem and LLPO, Appl. Math. Comput. 201 (2008) 378-383] Yasuhito Tanaka showed that Walras’ existence theorem implies the nonconstructive lesser limited principle of omniscience (LLPO); it follows that Walras’ existence theorem does not admit a constructive proof. We give a constructive proof of an approximate version of Walras’ existence theorem from which the full theorem can be recovered with an application of LLPO. We then push Uzawa’s equivalence theorem to the level of approximate solutions, before considering economies with at most one equilibrium.     

A Nash type solution for hemivariational inequality systems

In this paper, we prove an existence result for a general class of hemivariational inequality systems using the Ky Fan version of the KKM theorem Fan (1984) [10] or Tarafdar fixed points Tarafdar (1987) [11]. As application, we give an infinite-dimensional version for the existence result of Nash generalized derivative points introduced recently by Kristály (2010) [5]. We also give an application to a general hemivariational inequality system.     

On the existence of almost contact structure and the contact magnetic field

We give a simple proof of the existence of an almost contact metric structure on any orientable 3-dimensional Riemannian manifold (M ³, g) with the prescribed metric g as the adapted metric of the almost contact metric structure. By using the key formula for the structure tensor obtained in the proof this theorem, we give an application which allows us to completely determine the magnetic flow of the contact magnetic field in any 3-dimensional Sasakian manifold.     

A proof of the montague-plemmons-schein theorem on maximal subgroups of the semigroup of binary relations

The theorem in question is that the group of automorphisms of a partially ordered set (X,π), π denoting the order relation on the set X, is isomorphic to the maximal subgroup of ℬ_x containing π, where ℬ_x is the semigroup of all binary relations on X. This theorem is due to Montague and Plemmons [1] for the case X finite or countably infinite, and was extended by Schein to the general case, using a theorem due to Zaretsky [4]. A proof of the general case, based on [1] and results due to Plemmons and West [3], is also given in the preceding note by Plemmons and Schein [2]. The purpose of this note is to give an entirely self-contained proof of this intersesting theorem.     

General theorem for the existence of iterative roots of homeomorphisms with periodic points

We apply Zdun’s factorization theorem (see Zdun (2008) [3]) to give the conditions for the existence and the form of continuous and orientation-preserving iterative roots of homeomorphisms of the circle with a rational rotation number. Our theorem generalizes the previous results given by Jarczyk (2003) in [2], Zdun (2008) in [3] and Solarz (2003, 2009) in [4] and [5].     

Harmonic diffeomorphisms and Teichmüller theory

Teichmüller spaces of Riemann surfaces usually are treated by using quasi conformal mappings. We prove the existence of a harmonic diffeomorphism between punctured surfaces. We take this to build up a Teichmüller theory and give a new proof of Teichmüller's theorem for Riemann surfaces of finite type.     

A simple proof of the approximation by real analytic Lipschitz functions

A theorem in Azagra et al. (preprint) [1] asserts that on a real separable Banach space with separating polynomial every Lipschitz function can be uniformly approximated by real analytic Lipschitz function with a control over the Lipschitz constant. We give a simple proof of this theorem.