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We show that a compact Riemannian 3-manifold M with strictly convex simply connected boundary and sectional curvature K\leq a\leq 0 is isometric to a convex domain in a complete simply connected space of constant curvature a, provided that K\equiv a on planes tangent to the boundary of M. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard 3-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for K\geq a\geq 0. [ABSTRACT FROM AUTHOR]

Zhang (2021), Luo and Yin (2022) initiated the study of Lagrangian submanifolds satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂn or ℂℙn, where T = ∇ ∗ h ˜ and h ˜ is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂ2 under proper small energy assumption and gave new characterization of the Whitney spheres in ℂ2. In this paper, the authors extend these results to Lagrangian submanifolds in ℂn of dimension n ≥ 3 and to Lagrangian submanifolds in ℂℙn. [ABSTRACT FROM AUTHOR]

Zhang (2021), Luo and Yin (2022) initiated the study of Lagrangian submanifolds satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂn or ℂℙn, where T = ∇ ∗ h ˜ and h ˜ is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂ2 under proper small energy assumption and gave new characterization of the Whitney spheres in ℂ2. In this paper, the authors extend these results to Lagrangian submanifolds in ℂn of dimension n ≥ 3 and to Lagrangian submanifolds in ℂℙn. [ABSTRACT FROM AUTHOR]

In this paper, we prove some gap theorems for complete λ -hypersurfaces. Assume that the L n / 2 -norm of a quantity concerning the trace-free second fundamental form and the mean curvature is finite, we show that the hypersurface must be a plane. The same results for λ -surfaces in R 3 also given provided L 4 -curvature pinching conditions. [ABSTRACT FROM AUTHOR]

We consider the L 2 -energy functional E sym (f) of pullbacks of metrics by smooth maps f between Riemannian manifolds. Stationary maps for E sym (f) are called symphonic maps and researched in Kawai and Nakauchi (Nonlinear Anal 74:2284–2295, 2011; Differ Geom Appl 44:161–177, 2016; Differ Geom Appl 65:147–159, 2019), Misawa and Nakauchi (Nonlinear Anal 75:5971–5974, 2012; Calc Var Partial Differ Equ 55:1–20, 2016; Adv Differ Equ 23:693–724, 2018; Part Differ Equ Appl 2:19, 2021) and Nakauchi and Takenaka (Ricerche Matematica 60:219–235, 2011). In this paper we are concerned with the m-symphonic energy E sym m (f) , i.e., the L m 2 -version of the symphonic energy, where m denotes the dimension of the source manifold M. The m-symphonic energy is conformally invariant and is introduced in Misawa and Nakauchi (Part Differ Equ Appl 2:19, 2021). Under some conditions on this conformal energy E sym m (f) , we give two results—a gap theorem and a Liouville type theorem—for symphonic maps. [ABSTRACT FROM AUTHOR]

In this paper, we establish two gap theorems for ends of smooth metric measure space (M^n, g,e^{-f}dv) with the Bakry-Émery Ricci tensor \operatorname {Ric}_{f}\!\ge -(n-1) in a geodesic ball B_{o}(R) with radius R and center o\in M^n. When \operatorname {Ric}_{f}\ge 0 and f has some degeneration outside B_{o}(R), we show that there exists an \epsilon =\epsilon (n,\sup _{B_{o}(1)}|f|) such that such a space has at most two ends if R\le \epsilon. When \operatorname {Ric}_{f}\ge \frac 12 and f(x)\le \frac 14d^2(x,B_{o}(R))+c for some constant c>0 outside B_{o}(R), we can also get the same gap conclusion. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the gap phenomena for complete Lagrangian submanifolds satisfying ∇ ∗ T ≡ 0 in complex space forms N n (4 c) , where T = 1 n ∇ ∗ B and B is the Lagrangian trace-free second fundamental form. We prove that under some pointwise or integral curvature pinching condition, the submanifold is either totally geodesic or isometric to the Whitney sphere. [ABSTRACT FROM AUTHOR]

In this work, we consider M = (B r 3 , g ¯) as the Euclidean three-ball with radius r equipped with the metric g ¯ = e 2 h , conformal to the Euclidean metric, where the function h = h (x) depends only on the distance of x to the center of B r 3 . We show that if a free boundary CMC surface Σ in M satisfies a pinching condition on the length of the traceless second fundamental tensor which involves the support function of Σ , the positional conformal vector field x and its potential function σ , then either Σ is a disk or Σ is an annulus rotationally symmetric. These results extend to the CMC case and to many other different conformally Euclidean spaces the main result obtained by Li and Xiong (J Geom Anal 28(4):3171–3182, 2018). [ABSTRACT FROM AUTHOR]

In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with the L² norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the Lp norm for certain p > 2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere. [ABSTRACT FROM AUTHOR]

Abstract A classic gap theorem for complete non-compact Ricci flat manifolds by M. Anderson suggests that one can get the rigidity of certain spaces simply by passing local curvature estimates on geodesic balls to global. Using similar ideas, Kim showed that a 4-dimensional complete non-compact Bach-flat manifold with vanishing scalar curvature and small L 2 -curvature tensor has to be flat. Unfortunately, this method does not generalize to compact manifolds without boundary. Applying a different approach, we show that a closed Bach-flat Riemannian manifold with positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either L ∞ or L n 2 -norm. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A. Singer. As an application, we can partially recover the well-known Chang–Gursky–Yang's 4-dimensional conformal sphere theorem. [ABSTRACT FROM AUTHOR]

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere. [ABSTRACT FROM AUTHOR]

In this short note, using Günther’s volume comparison theorem and Yokota’s gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton (Mn, g,f) with sectional curvature K(g) < A and Volf (M) ≥ v for some uniform constant A, v, there exists a small uniform constant εn,A,v > 0 depends only on n,A and v, if the scalar curvature R ≤ εn,A,v, then (M,g, f) is isometric to the Gaussian soliton (ℝn, gE, |x|2/ 4 ). [ABSTRACT FROM AUTHOR]

We study, from the extrinsic point of view, the structure at infinity of open submanifolds, $$\varphi :M^m \hookrightarrow \mathbb {M}^{n}(\kappa )$$ isometrically immersed in the real space forms of constant sectional curvature $$\kappa \le 0$$ . We shall use the decay of the second fundamental form of the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in Greene et al. (Int Math Res Not 1994:364-377, 1994) and Petrunin and Tuschmann (Math Ann 321:775-788, 2001) and 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. [ABSTRACT FROM AUTHOR]

A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, by applying a linear trace Li-Yau-Hamilton inequality for a positive (1, 1)-form solution of the CR Hodge-Laplace heat equation and monotonicity of the heat equation deformation, we obtain an optimal gap theorem for a complete strictly pseudoconvex CR (2n + 1)-manifold with nonnegative pseudohermitian bisectional curvature and vanishing torsion. We prove that if the average of the Tanaka-Webster scalar curvature over a ball of radius r centered at some point o decays as o (r-2), then the manifold is flat. [ABSTRACT FROM AUTHOR]

In this paper, using estimates for the generalized Ricci curvature, we shall give some gap theorems for Ricci-harmonic solitons with compact domain manifolds by showing some necessary and sufficient conditions for the solitons to be harmonic-Einstein. Our results may be regarded as generalizations of recent works by H. Li, and M. Fernández-López and E. García-Río. [ABSTRACT FROM AUTHOR]

In this paper, we prove a gap result for a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative Ricci curvature and a scalar curvature average condition. We show that if it has positive Green function, then it is flat. This result is proved by setting up new global Yamabe flow. Other extensions related to bounded positive solutions to a Schrodinger equation are also discussed. A global existence of Yamabe flow on a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative sectional curvature is proved. [ABSTRACT FROM AUTHOR]

We study a complete noncompact hypersurface M n isometrically immersed in an ( n + 1 ) -dimensional sphere S n + 1 ( n ≥ 3 ) . We prove that there is no non-trivial L 2 -harmonic 2-form on M , if the length of the second fundamental form is less than a fixed constant. We also showed that the same conclusion holds if the scale-invariant total tracefree curvature is bounded above by a small constant depending only on n . These results are generalized versions of the result of Cheng and Zhou on bounded harmonic functions with finite Dirichlet integral and the one of Fang and the author on L 2 harmonic 1-forms. [ABSTRACT FROM AUTHOR]

In this paper, by using estimates for the transverse Ricci curvature in terms of the Sasaki-Futaki invariant, we shall give some gap theorems for compact gradient Sasaki-Ricci solitons by showing some necessary and sufficient conditions for the solitons to be Sasaki-Einstein. Our results may be regarded as a Sasaki geometry version of recent works by Li and Fernández-López and García-Río. [ABSTRACT FROM AUTHOR]

Given a complete isometric immersion φ: P⟶ N in an ambient Riemannian manifold N with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space $M^{n}_{w}$, we determine a set of conditions on the extrinsic curvatures of P that guarantee that the immersion is proper and that P has finite topology in line with the results reported in Bessa et al. (Commun. Anal. Geom. 15(4):725-732, ) and Bessa and Costa (Glasg. Math. J. 51:669-680, ). When the ambient manifold is a radially symmetric space, an inequality is shown between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends, which generalizes the classical inequality stated in Anderson (Preprint IHES, ) for complete and minimal submanifolds in ℝ. As a corollary we obtain the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in hyperbolic space, together with Bernstein-type results for such submanifolds in Euclidean and hyperbolic spaces, in the manner of the work Kasue and Sugahara (Osaka J. Math. 24:679-704, ). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove a gap theorem for translating solitons to the almost calibrated Lagrangian mean curvature flow. More precisely, we prove that there exists a constant such that if , then the translating soliton must be a plane. We also obtain a similar result for symplectic mean curvature flow. [Copyright &y& Elsevier]

Necessary and sufficient conditions for a gradient Ricci soliton to be Einstein are given, showing that they can be expressed in terms of upper and lower bounds on the behavior of the Ricci tensor when evaluated on the gradient of the potential function of the soliton. [ABSTRACT FROM AUTHOR]

Abstract: Symmetric disk graphs are often used to model wireless communication networks. Given a set S of n points in (representing n transceivers) and a transmission range assignment , the symmetric disk graph of S (denoted ) is the undirected graph over S whose set of edges is , where denotes the Euclidean distance between points u and v. We prove that the weight of the MST of any connected symmetric disk graph over a set S of n points in the plane, is only times the weight of the MST of the complete Euclidean graph over S. We then show that this bound is tight, even for points on a line. Next, we prove that if the number of different ranges assigned to the points of S is only k, , then the weight of the MST of is at most 2k times the weight of the MST of the complete Euclidean graph. Moreover, in this case, the MST of can be computed efficiently in time . We also present two applications of our main theorem, including an alternative proof of the Gap Theorem, and a result concerning range assignment in wireless networks. Finally, we show that in the non-symmetric model (where ), the weight of a minimum spanning subgraph might be as big as times the weight of the MST of the complete Euclidean graph. [Copyright &y& Elsevier]

Thanks to Trachtenbrot and Borodin, complexity theorists have the Gap Theorem, which guarantees the existence of arbitrarily large gaps in the complexity hierarchy. That is, there can be made arbitrarily large increases in resource availability that, for all their vastness, yield no additional computational power: every computation that can be performed given the extra resource could have been performed in its absence. In response to needs arising during the complexity analysis of unconventional-quantum, analogue, chemical, optical, etc.--computers, the notion of dominance was introduced (we recap in the present paper the relevant definitions). With dominance comes a corresponding hierarchy of complexity classes, and it is natural to ask whether a gap theorem, analogous to that of Trachtenbrot and Borodin, holds in this context. We show that a gap theorem does indeed hold for certain dominance-based classes. We consider related statements concerning other dominance-based classes, and formulate corresponding conjectures. [ABSTRACT FROM AUTHOR]

Abstract: Consider a local chain Differential Graded algebra, such as the singular chain complex of a pathwise connected topological group. In two previous papers, a number of homological results were proved for such an algebra: An Amplitude Inequality, an Auslander–Buchsbaum Equality, and a Gap Theorem. These were inspired by homological ring theory. By the so-called looking glass principle, one would expect that analogous results exist for simply connected cochain Differential Graded algebras, such as the singular cochain complex of a simply connected topological space. Indeed, this paper establishes such analogous results. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]