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We obtain a sharp estimate to the norm of the traceless second fundamental form of complete hypersurfaces with constant scalar curvature immersed into a locally symmetric Riemannian manifold obeying standard curvature constraints (which includes, in particular, the Riemannian space forms with constant sectional curvature). When the equality holds, we prove that these hypersurfaces must be isoparametric with two distinct principal curvatures. Our approach involves a suitable Okumura type inequality which was introduced by Melendez (Bull Braz Math Soc 45:385–404, 2014) , corresponding to a weaker hypothesis when compared with to the assumption that these hypersurfaces have a priori at most two distinct principal curvatures.

The Three Gap Theorem states that there are at most three distinct lengths of gaps if one places $n$ points on a circle, at angles of $z, 2z, 3z, \ldots nz$ from the starting point. The theorem was first proven in 1958 by S\'os and many proofs have been found since then. In this note we show how the Three Gap Theorem can easily be deduced by using Rauzy-Veech induction., Comment: 5 pages

In this paper, inspired by Fernandez-Lopez and Garcia-Rio [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.

Let f : ( Y , g ) → ( X , g 0 ) be a nonzero degree continuous map between compact Kahler manifolds of dimension n ≥ 2 , where g 0 has constant negative holomorphic sectional curvature. Adapting the Besson–Courtois–Gallot barycentre map techniques to the Kahler setting, we prove a gap theorem in terms of the degree of f and the diastatic entropies of ( Y , g ) and ( X , g 0 ) which extends the rigidity result proved by the author in [13] .

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni, An optimal gap theorem, Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and the first author [L. Ni and L.-F. Tam, Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature, J. Differential Geom. 64 2003, 3, 457–524] and complements a recent result of Liu [G. Liu, Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds, Duke Math. J. 165 2016, 15, 2899–2919].

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in $R^{n+k}$, namely, if the $L^n$ norm of the second fundamental form of the soliton is small enough, then it is a hyperplane., Comment: some results are reformulated and the monotonicity formula and volume growth are added

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space. The above curvature estimates are proved using a gap theorem for Ricci-flatness on homogeneous spaces. The proof of this gap theorem is by contradiction and uses a local $W^{2,p}$ convergence result, which holds without symmetry assumptions., Fixed up the formulation of the weak convergence theorem in Section 2

Motivated by a previous work by Zheng and the second-named author, we study pinching constants of compact Kahler manifolds with positive holomorphic sectional curvature. In particular, we prove a gap theorem on Kahler manifolds with almost-one-half pinched holomophic sectional curvature. The proof is motivated by the work of Petersen and Tao on Riemannian manifolds with almost-quarter-pinched sectional curvature.

In the framework of effectively enumerable topological spaces, we introduce the notion of a partial computable function. We show that the class of partial computable functions is closed under composition, and the real-valued partial computable functions defined on a computable Polish space have a principal computable numbering. With respect to the principal computable numbering of the real-valued partial computable functions, we investigate complexity of important problems such as totality and root verification. It turns out that for some problems the corresponding complexity does not depend on the choice of a computable Polish space, whereas for other ones the corresponding choice plays a crucial role.

Usually, the Weierstras gap theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstras gap theorem by following an alternative approach based on “first principles”, which does not use the Riemann– Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface is associated with the modular curve X0(N).

We show that a compact half-conformally flat manifold of negative type with bounded L2 energy, sufficiently small scalar curvature, and a noncollapsing assumption has all Betti numbers bounded in terms of the L2 curvature norm. We give examples of multi-ended bubbles that disrupt attempts to improve these Betti number bounds. We show that bounded self-dual solutions of dω=0 on asymptotically locally Euclidian (ALE) manifold ends display a rate-of-decay gap: they are either asymptotically Kahler, or they have a decay rate of O(r−4) or better.

We show that there exists a positive real number $\delta>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-\delta$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above., Comment: 39 pages, comments are welcome; v2: 40 pages, slightly modified, more discussion on computation of $\delta$ added; v3: final version, to appear in J. Algebraic Geom

An old classical result in computable structure theory is Ash’s theorem stating that for every computable ordinal α ≥ 2, under some additional conditions, a computable structure is Δ 0 -categorical iff it has a computable Σ α Scott family. We construct a counterexample revealing that the proof of this theorem has a serious error. Moreover, we show how the error can be corrected by revising the proof. In addition, we formulate a sufficient condition under which the Δ 0 -dimension of a computable structure is infinite.

Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose first order terms have coefficients with compact support in the open domain. The eigenvalue gap is bounded below by the gap of an associated Sturm-Liouville problem on a closed interval.

The three gap theorem, also known as the Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of $\alpha,2\alpha,\ldots, N\alpha$ take at most three distinct values. Motivated by a question of Erd\H{o}s, Geelen and Simpson, we explore a higher-dimensional variant, which asks for the number of gaps between the fractional parts of a linear form. Using the ergodic properties of the diagonal action on the space of lattices, we prove that for almost all parameter values the number of distinct gaps in the higher dimensional problem is unbounded. Our results in particular improve earlier work by Boshernitzan, Dyson and Bleher et al. We furthermore discuss a close link with the Littlewood conjecture in multiplicative Diophantine approximation. Finally, we also demonstrate how our methods can be adapted to obtain similar results for gaps between return times of translations to shrinking regions on higher dimensional tori., Comment: 19 pages

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of the relative volume inequality $(1.9)$ on conformally compact Einstein manifolds. Therefore we obtain the complete proof of the rigidity theorem for conformally compact Einstein manifolds in general dimensions with no spin structure assumption (cf. $[29, 15]$) as well as the new curvature pinch estimates for conformally compact Einstein manifolds with conformal infinities of very large Yamabe constant. We also derive the curvature estimates for conformally compact Einstein manifolds with conformal infinities of large Yamabe constant., Comment: 28 pages, 1 figure(with one sentence added)

We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature $\kappa \leq 0$. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in the papers Internat. Math. Res. Notices 1994, no. 9, authored by R. E. Greene, P. Petersen and S. Zhou and Math. Ann. 2001, 321 (4), authored by A. Petrunin and W. Tuschmann. We shall obtain too an estimation from below of the number of its ends in terms of the volume growth of a special class of extrinsic domains, the extrinsic balls., Comment: 19 pages

In an attempt to answer the question raised by A.W. Goodman, we obtain a covering theorem, a distortion theorem, a growth theorem, the radius of convexity and an argument estimate of f ′ ( z ) for functions of the class σ of bi-univalent functions.

We study a complete noncompact submanifold M n in a sphere S n + p . We prove that there admit no nontrivial L 2 -harmonic 1-forms on M if the total curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carronʼs, Yunʼs, Cavalcanteʼs and the first authorʼs results on submanifolds in Euclidean spaces and Seoʼs result on submanifolds in hyperbolic space without the condition of minimality.

Pursuing the study started by Rigo, Salimov and Vandomme, we use elementary number-theoretic techniques to characterize rotation words having a finite set of abelian returns to all prefixes. We also make the connection between the three gap theorem and the number of semi-abelian returns for Sturmian words, simplifying some arguments developed by Puzynina and Zamboni.

We define the effective integrability of Fine-computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integrals such as the Bounded Convergence Theorem, the Dominated Convergence Theorem, and the Second Mean Value Theorem. It is also proved that the Walsh-Fourier coefficients of an effectively integrable Fine-computable function form a Euclidian computable sequence of reals which converges effectively to zero. This property of convergence is the effectivization of the Walsh-Riemann-Lebesgue Theorem. The article is closed with the effective version of Dirichlet's test. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this article, we study various homological invariants of differential graded (DG for short) modules over a connected DG algebra following Frankild–Jorgensen. Two different versions of homological dimensions (resolutional and functorial) are defined. In some cases, they are proved to be simply the bound of the cohomology of the DG module. Some homological identities, such as Auslander–Buchsbaum formula and Bass formula, are proved for compact DG modules over a connected DG algebra.

We consider the distribution of the maximum for finite, deterministic, periodic and quasiperiodic sequences, and contrast the extreme value distributions in these cases with the classical results for iidrv's. A significant feature in the case of deterministic sequences is a multi-step structure for the distribution function. The extreme value distribution for the circle map with an irrational parameter is obtained in closed form with the help of the three-gap theorem for the map Xj+1 = (Xj + a) mod 1 where a ∈ (0,1) is an irrational number.

We define the effective integrability of Fine-computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integral such as Bounded Convergence Theorem and Dominated Convergence Theorem. It is also proved that the Walsh-Fourier coefficients of an effectively integrable Fine-computable function form an E-computable sequence of reals and converge effectively to zero. The latter fact is the effectivization of Walsh-Riemann-Lebesgue Theorem. The article is closed with the effective version of Dirichlet's test.

We give a rigidity theorem of proper holomorphic mappings between generalized pseudoellipsoids. The theorem claims that any proper holomorphic mapping which is holomorphic extendable up to the boundary between generalized pseudoellipsoids of non-equidimensions is a collections of totally geodesic embeddings up to automorphisms.

We present computable versions of the Fréchet–Riesz Representation Theorem and the Lax–Milgram Theorem. The classical versions of these theorems play important roles in various problems of mathematical analysis, including boundary value problems of elliptic equations. We demonstrate how their computable versions yield computable solutions of the Neumann and Dirichlet boundary value problems for a simple non-symmetric elliptic differential equation in the one-dimensional case. For the discussion of these elementary boundary value problems, we also provide a computable version of the Theorem of Schauder, which shows that the adjoint of a computably compact operator on Hilbert spaces is computably compact again.

In this article we give an extension of a theorem of N. Levinson (see Theorem 2.1). As an application, we get a variation of the Fabry Gap theorem (see Theorem 2.3) concerning the location of singularities of Taylor–Dirichlet series, on the boundary of convergence.

Given an orientation-preserving diffeomorphism of the interval [0;1], consider the uniform norm of the differential of its n-th iteration. We get a function of n called the growth sequence. Its asymptotic behaviour is an interesting invariant which naturally appears both in geometry of the diffeomorphisms groups and in smooth dynamics. Our main result is the following Gap Theorem: the growth rate of this sequence is either exponential, or at most quadratic with n. Further, we construct diffeomorphisms with quite an irregular behaviour of the growth sequence. This construction easily extends to arbitrary manifolds., Comment: LATeX, 24 pp, incorporates DS/0110306, the latest version

We investigate the effects of adding uniformity requirements to concepts in computable structure theory such as computable categoricity (of a structure) and intrinsic computability (of a relation on a computable structure). We consider and compare two different notions of uniformity, previously studied by Kudinov and by Ventsov. We discuss some of their results and establish new ones, while also exploring the connections with the relative computable structure theory of Ash, Knight, Manasse, and Slaman and Chisholm and with previous work of Ash, Knight, and Slaman on uniformity in a general computable structure-theoretical setting.

We give a Volterra-series representation theorem for discrete-space input-output maps that are analytic. The theorem provides a bound on the error in approximating the series with its first p + 1 terms. Examples are given to illustrate how the theorem can be used.

We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally we apply these results to prove a gap theorem for power series which are remainders of the Grauert-Hironaka-Galligo Division Theorem., Comment: Final version - 48 pp - to appear in J. Reine Angew. Math

We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other.

Here we shall use the same notations of the original paper. The references to equations, lemmas, propositions, theorems are with respect to the paper unless stated otherwise. The argument provided for the proof of Proposition 3.1 is not complete. It overlooks the fact that the solution 2-form η(x, t) of the problem (3.1) may contain non-trivial (2,0) and (0,2) parts. Hence Proposition 3.1 needs a correction. We thank Gilles Carron for pointing out this oversight in the original proof of Proposition 3.1 and his comments. We also thank Luen-Fai Tam for helpful related discussions. We refer interested reader to [5] for an alternate proof of Theorem 1.2. Since the preservation of the positivity for solution 2-forms solving the problem (3.1) in the paper, without knowing that the 2-form being (1,1)-type, seems elusive (even for the (1,1) part projection of the 2-form solutions) we replace (3.1) with the following initial boundary value problem on (1,1)form η. Let Ω be a smooth bounded domain in M . For a given point p ∈ ∂Ω , let ν be the unit outward normal in a small neighborhood near p. We

(2006). A Simple Way of Proving the Jordan-Holder-Schreier Theorem. The American Mathematical Monthly: Vol. 113, No. 10, pp. 933-935.

Here Sr , r = 1, . . . , n, is the r th-symmetric function of the principal curvatures of the immersion and L1 is the linearized operator corresponding to the equation of hypersurfaces of Sn+1 with S2 ≡ 0. The proof of Lemma 4.1 presented in [AdCS] is incorrect. We had set N = ∑n+2 i=1 niei , where {e1, . . . , en+2} is an orthonormal basis of Rn+2 and N is the unit normal vector to the immersion of M → Sn+1 ⊂ Rn+2. By assuming the index of I is greater than one and noticing that I (ni, ni) ≤ 0, we concluded that all ni but one were zero. This is not true. Replace the proof of the lemma by the following.

In [B1] the first author proved a theorem about differential algebraic groups (in the sense of Cassidy-Kolchin [C], [K2]) that implied the geometric LangMordell conjecture [L]; cf. Theorem 1 below. The proof in [B1] used an analytic argument, based on “Big Picard”. Hrushovski [H] was able to replace this analytic argument by a remarkable model theoretic argument, based in its turn on the difficult theory developed in [HZ]. Now Hrushovski’s model theoretic methods further lead to striking new results on differential algebraic groups [HS]; cf. Theorem 3 below. So a natural challenge presents itself: to find a proof for these new results that is free from model theory. In this note we shall give in particular a quick analytic proof of Theorem 3, based, again, on “Big Picard”. The main result of this note is the “Gap” Theorem 2 below, which should be viewed as a significant complement to Theorem 1; Theorems 1 and 2 will then easily imply Theorem 3. Some comments are in order concerning the role of the authors in this paper. The second author asked whether analytic methods as in [B1] could yield strong minimality and local modularity of the “Manin kernel” of a simple abelian variety which does not descend to the constants (Theorem 3 below). The first author then proved strong minimality (assertion 1 of Theorem 3) by proving a weaker version of Theorem 2 (namely for simple abelian varieties). The second author suggested some general ideas, analogous to those in [P1], for obtaining local modularity. In trying to carry out details, the first author came up with the proof of the present Theorem 2, from which everything follows. Recall some basic terminology of differential algebra [K1], [C], [B2]. (The definitions below will suffice to understand the statements of the Theorems below, without assuming any previous knowledge of differential algebra; for the proofs, however, familiarity with [B1], [B2] is required.) Let F be a δ−field (i.e. a field of characteristic zero equipped with a derivation δ.) One defines the ring of δ−polynomials F{y1, ..., yN} as the ring of usual polynomials with F−coefficients in the variables δyj , 1 ≤ j ≤ N, i ≥ 0. There is an obvious notion of order for δ−polynomials. F is said to be δ−closed if for any A, B ∈ F{y}, B = 0, such that the order of A is strictly bigger than the order of B there

In this paper, we develop a method of solving the Poincaré–Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge–Laplace heat equation on$(1, 1)$-forms. The method is effective in proving an optimal result when$M$has nonnegative bisectional curvature. It also provides an alternate proof of a recent gap theorem of the first author.

In 1965 Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in $R^3$ is at least $2\pi^2$ and attains this minimal value if and only if the torus is a M\"obius transform of the Clifford torus. This was recently proved by Marques and Neves. In this paper, we show for tori there is a gap to the next critical point of the Willmore energy and we discuss an application to the Willmore flow. We also prove an energy gap from the Clifford torus to surfaces of higher genus., Comment: 9 pages. In this new version we performed some small changes to improve the exposition. To appear in Nonlinear Analysis: Theory Methods & Applications

We consider critical points of the functionals $\Pi$ and $\Psi$ defined as the global $L^2$-norm of the second fundamental form and mean curvature vector of isometric immersions of compact Riemannian manifolds into a background Riemannian manifold, respectively, as functionals over the space of deformations of the immersion. We prove gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the sphere can be cast as a theorem about critical points of these functionals of constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various type of minimal submanifolds that can occur at the pointwise upper bound on the norm of the second fundamental form are realized by manifolds of nonnegative Ricci curvature, and of these, the Einstein ones are distinguished from the others by being those that are immersed on the sphere as critical points of $\Pi$., Comment: This version corrects a simple notational error in referring to the norm of A_{\nu_H}

The questions considered in this paper arose from the study [KS] of I. Fredholm's (insufficient) proof that the gap series S08 an ?n2 (where 0 < |a| < 1) is nowhere continuable across {|?| = 1}. The interest of Fredholm's method ([F],[ML]) is not so much its efficacy in proving gap theorems (indeed, much more general results can be got by other means, cf. the Fabry gap theorem in [Di]) as in the connection it made between certain special gap series and partial differential equations (...).

Following [T. Colding and W. Minicozzi, II, Generic mean curvature flow I; generic singularities, Ann. of Math. 175(2) (2012) 755–833], we define a notion of entropy for connections over [Formula: see text] which has shrinking Yang–Mills solitons as critical points. As in [T. Colding and W. Minicozzi, II, Generic mean curvature flow I; generic singularities, Ann. of Math. 175(2) (2012) 755–833], this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying “generic singularities” of the Yang–Mills flow, and we discuss the differences in this strategy in dimension [Formula: see text] versus [Formula: see text].

23 p. -- An extended abstract of this work appears in the proceedings of the 2012 ACM/IEEE Symposium on Logic in Computer Science We prove an algebraic preservation theorem for positive Horn definability in aleph-zero categorical structures. In particular, we define and study a construction which we call the periodic power of a structure, and define a periomorphism of a structure to be a homomorphism from the periodic power of the structure to the structure itself. Our preservation theorem states that, over an aleph-zero categorical structure, a relation is positive Horn definable if and only if it is preserved by all periomorphisms of the structure. We give applications of this theorem, including a new proof of the known complexity classification of quantified constraint satisfaction on equality templates. The first author was partially supported by the Spanish program "Ramon y Cajal" and MICINN grant TIN2010-20967-C04-02. The first author was also supported by by Spanish Project FORMALISM(TIN2007-66523), by the Basque Government Project S-PE12UN050(SAI12/219), and by the University of the Basque Country under grant UFI11/45. The second author thanks the FWF (Austrian Science Fund) for its support through Project P 24654 N25.

The Whitney-Mahowald theorem gave normal Euler number ( mod 4 ) (\bmod \,4) for embeddings of a closed 2 n 2n -manifold in Euclidean 4 n 4n -space. We generalize this theorem to embeddings of closed 2 n 2n -manifolds in an oriented 4 n 4n -manifold with an approach in the framework of unoriented bordism groups of maps.