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Our main goal in this work is to deal with results concern to the σ2-curvature. First we find a symmetric 2-tensor canonically associated to the σ2-curvature and we present an almost-Schur-type lemma. Using this tensor, we introduce the notion of σ2-singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and σ2-curvature. With a suitable condition on the σ2-curvature, we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the three-dimensional torus does not admit a metric with constant scalar curvature and nonnegative σ2-curvature unless it is flat. [ABSTRACT FROM AUTHOR]

Abstract: Let G2(2) be the non‐compact connected simple Lie group of type G2 over R, and let M be a connected analytic complete pseudo‐Riemannian manifold that admits an isometric G2(2)‐action with a dense orbit. For the case dim ( M ) ≤ 21, we provide a full description of the manifold M, its geometry and its G2(2)‐action. The latter are always given in terms of a Lie group geometry related to G2(2), and in one case M is essentially the quotient of SO0(3, 4) by a lattice. [ABSTRACT FROM AUTHOR]

We prove the existence of manifolds with almost maximal volume entropy which are not hyperbolic. [ABSTRACT FROM AUTHOR]

The paper is based on the lecture course “Metric projective geometry” which I conducted at the summer school “Finsler geometry with applications” at Karlovassi, Samos, in 2014, and at the workshop before the 8th seminar on Geometry and Topology of the Iranian Mathematical society at the Amirkabir University of Technology in 2015. The goal of this lecture course was to show how effective projectively invariant objects can be used to solve natural and named problems in differential geometry, and this paper also does it: I give easy new proofs to many known statements and also prove the following new statement: on a complete Riemannian manifold on nonconstant curvature, the index of the group of affine transformations in the group of projective transformations is at most two. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds M3 with nonnegative Ricci curvature and strictly convex boundary ∂M. Here we obtain a sharp upper bound for the length L(∂Σ) of the boundary ∂Σ of a free boundary minimal surface Σ2 in M3 in terms of the genus of Σ and the number of connected components of ∂Σ, assuming Σ has index one. After, under a natural hypothesis on the geometry of M along ∂M, we prove that if L(∂Σ) saturates the respective upper bound, then M3 is isometric to the Euclidean 3-ball and Σ2 is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of Σ, when M3 is a strictly convex body in R3, which is saturated only on the Euclidean 3-balls (by the Euclidean disks). We also consider similar results for free boundary stable CMC surfaces. [ABSTRACT FROM AUTHOR]

If a graph submanifold (x, f(x)) of a Riemannian warped product space (Mm×eψNn,g~=g+e2ψh) is immersed with parallel mean curvature H, then we obtain a Heinz-type estimation of the mean curvature. Namely, on each compact domain D of M, m‖H‖≤Aψ(∂D)Vψ(D) holds, where Aψ(∂D) and Vψ(D) are the ψ-weighted area and volume, respectively. In particular, H=0 if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.01257, 2016). This generalizes the known cases n=1 or ψ=0. We also conclude minimality using a closed calibration, assuming (M,g∗) is complete where g∗=g+e2ψf∗h, and for some constants α≥δ≥0, C1>0 and β∈[0,1), ‖∇∗ψ‖g∗2≤δ, Ricciψ,g∗≥α, and detg(g∗)≤C1r2β holds when r→+∞, where r(x) is the distance function on (M,g∗) from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field. [ABSTRACT FROM AUTHOR]

Abstract: Let ( M , g ) be an n -dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying ρ Δ g ρ ⩾ n − 5 ⩾ 0, where Δ g is the Laplace–Beltrami operator on ( M , g ) and ρ is the distance function from a given point. If ( M , g ) supports a second‐order Sobolev inequality with a constant C > 0 close to the optimal constant K 0 in the second‐order Sobolev inequality in R n, we show that a global volume noncollapsing property holds on ( M , g ). The latter property together with a Perelman‐type construction established by Munn (J. Geom. Anal. (2010) 723–750) provide several rigidity results in terms of the higher order homotopy groups of ( M , g ). Furthermore, it turns out that ( M , g ) supports the second‐order Sobolev inequality with the constant C = K 0 if and only if ( M , g ) is isometric to the Euclidean space R n. [ABSTRACT FROM AUTHOR]

We obtain some De Lellis-Topping type inequalities on the smooth metric measure spaces, some of them are as generalization of De Lellis-Topping type inequality that was proved by X. Cheng [Ann. Global Anal. Geom., 2013, 43: 153-160]. [ABSTRACT FROM AUTHOR]

We study special flows associated to Hitchin representations of a surface group, namely the simple root flows. We introduce and discuss the Liouvlle geodesic current which plays a singular role amongst all the natural invariant currents associated to such a representation. Finally, we discuss a rigidity result and a pressure metric associated to the first simple root flow. [ABSTRACT FROM AUTHOR]

Let ( M n , g , X ) be a complete generic shrinking Ricci soliton of dimension n ≥ 3 . In this paper, by employing curvature inequalities, the formula of X -Laplacian for the norm square of the trace-free curvature tensor, the weak maximum principle and the estimate of the scalar curvature of ( M n , g ) , we prove some rigidity results for ( M n , g , X ) . In particular, it is showed that ( M n , g , X ) is isometric to R n or a finite quotient of S n under a pointwise pinching condition. Moreover, we establish several optimal inequalities and classify those shrinking solitons for equalities. [ABSTRACT FROM AUTHOR]

There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. By means of a purely synthetic technique, we get a rigidity result for the sphere without any curvature conditions, completeness or compactness, as well as a dual result for the hyperbolic plane, the spacelike sphere in the Minkowski space. [ABSTRACT FROM AUTHOR]

Let M be an n-dimensional oriented compact submanifold with parallel mean curvature in the unit sphere $$S^{n+p}$$ . Denote by H and S the mean curvature and the squared length of the second fundamental form of M, respectively. We obtain a classification theorem of M if it satisfies $$S+\lambda _2\le \alpha (n,H)$$ , where $$\lambda _{2}$$ is the second largest eigenvalue of the fundamental matrix and $$\alpha (n,H)$$ is defined as in Theorem B. [ABSTRACT FROM AUTHOR]

Using the generalized Yau parameter method and the Sylvester theory, we verify that if M is a compact minimal hypersurface in S n + 1 whose squared length of the second fundamental form satisfies 0 ≤ | A | 2 − n ≤ n 22 , then | A | 2 ≡ n and M is a Clifford torus. Moreover, we prove that if M is a complete self-shrinker with polynomial volume growth in R n + 1 , and if the squared length of the second fundamental form of M satisfies 0 ≤ | A | 2 − 1 ≤ 1 21 , then | A | 2 ≡ 1 and M is a round sphere or a cylinder. Our results improve the rigidity theorems due to Ding and Xin [21,22] . [ABSTRACT FROM AUTHOR]

Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It is presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show that the hyperplane is rigid in an even stronger sense, namely: for $$2\le n \le 6$$ , any smooth, complete self-shrinker $$\Sigma ^n\subset \mathbf {R}^{n+1}$$ that is graphical inside a large, but compact, set must be a hyperplane. In fact, this rigidity holds within a larger class of almost stable self-shrinkers. A key component of this paper is the procurement of linear curvature estimates for almost stable shrinkers, and it is this step that is responsible for the restriction on n. Our methods also yield uniform curvature bounds for translating solitons of the mean curvature flow. [ABSTRACT FROM AUTHOR]

Given a Riemannian spin $$^c$$ manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O'Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kähler-Einstein manifold or a Riemannian product of a Kähler-Einstein manifold with $$\mathbb R$$ (or with the circle $$\mathbb S^1$$ ). [ABSTRACT FROM AUTHOR]

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an application, we show that the Thurston asymmetric metric is well-defined on the Teichmüller space of hyperbolic cone surfaces with fixed cone angles and boundary lengths. We compare such a Teichmüller space with the Teichmüller space of complete hyperbolic surfaces with punctures, by showing that the two spaces (endowed with the Thurston metric) are almost isometric. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate complete generic shrinking Ricci solitons with pointwise pinched curvature and prove two classification results for such manifolds. In particular, we show that any n -dimensional generic shrinking Ricci soliton ( M n , g , X ) is isometric to a finite quotient of R n or S n under some pointwise pinched curvature condition. The arguments mainly rely on the Δ X -type parabolicity of ( M n , g , X ) and algebraic curvature estimates. [ABSTRACT FROM AUTHOR]

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and the Chern-Moser tensor of the Sasakian pseudo-Einstein manifolds, respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian space forms among Sasakian pseudo-Einstein manifolds, respectively. [ABSTRACT FROM AUTHOR]

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an ( n + k)-dimensional complete Riemannian manifold $$\overline M $$ of non-negative ( n−1)-th Ricci curvature. The Liouville type theorem about the p-harmonic map with finite L -energy from complete submanifold in a partially non-negatively curved manifold to non-positively curved manifold is also obtained. [ABSTRACT FROM AUTHOR]

As a counterpart of the classical Yamabe problem, a fractional Yamabe flow has been introduced by Jin and Xiong (2014) on the sphere. Here we pursue its study in the context of general compact smooth manifolds with positive fractional curvature. First, we prove that the flow is locally well posed in the weak sense on any compact manifold. If the manifold is locally conformally flat with positive Yamabe invariant, we also prove that the flow is smooth and converges to a constant fractional curvature metric. We provide different proofs using extension properties introduced by Chang and González (2011) for the conformally covariant fractional order operators. [ABSTRACT FROM PUBLISHER]

The classical Liouville Theorem on conformal transformations determines local conformal transformations on the Euclidean space of dimension $${\ge }3$$ . Its natural adaptation to the general framework of Riemannian structures is the 2-rigidity of conformal transformations, that is such a transformation is fully determined by its 2-jet at any point. We prove here a similar rigidity for generalized conformal structures defined by giving a one parameter family of metrics (instead of scalar multiples of a given one) on each tangent space. [ABSTRACT FROM AUTHOR]

In this paper, we study the connectedness at infinity of complete submanifolds by using the theory of p-harmonic function. For lower-dimensional cases, we prove that if M is a complete orientable noncompact hypersurface in $$\mathbb {R}^{n+1}$$ and if $$\delta $$ -stability inequality holds on M, then M has only one p-nonparabolic end. It is also proved that if $$M^n$$ is a complete noncompact submanifold in $${\mathbb {R}}^{n+k}$$ with sufficiently small $$L^n$$ norm of the traceless second fundamental form, then M has only one p-nonparabolic end. Moreover, we obtain a lower bound of the fundamental tone of the p Laplace operator on complete submanifolds in a Riemannian manifold. [ABSTRACT FROM AUTHOR]

We classify isoparametric hypersurfaces in complex hyperbolic spaces. [ABSTRACT FROM AUTHOR]

We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any n -dimensional ( n ≥ 4 ) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite quotient of N n − k × R k , ( k > 0 ) , the product of a Einstein manifold N n − k with the Gaussian shrinking soliton R k . The technique applies also to the steady and expanding cases in all dimensions. In particular, we prove that a three dimensional gradient steady soliton with third order divergence-free Cotton tensor, i.e. with vanishing double divergence of the Bach tensor, is either flat or isometric to the Bryant soliton. [ABSTRACT FROM AUTHOR]

We prove that an n-dimensional, $$n\ge 4$$ , compact gradient shrinking Ricci soliton satisfying a $$L^{\frac{n}{2}}$$ -pinching condition is isometric to a quotient of the round $$\mathbb {S}^n$$ , which improves the rigidity theorem given by Catino (Integral pinched shrinking Ricci solitons, 2016), in dimension $$4\le n\le 6$$ . [ABSTRACT FROM AUTHOR]

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogeneous. [ABSTRACT FROM AUTHOR]

This article is an attempt to understand harmonic and holomorphic maps between two bounded symmetric domains in special situations. We study foliations associated to a lattice-equivariant harmonic map of small rank from a complex ball to another. The result is related to rigidity of some complex two ball quotients. Some open questions are raised as well. [ABSTRACT FROM AUTHOR]

Let M be an m-dimensional manifold and A = D / I = R⊕ N a Weil algebra of height r. We prove that any A-covelocity T f ∈ T * M, x ∈ M is determined by its values over arbitrary max{width A, m} regular and under the first jet projection linearly independent elements of T M. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result T * M ≃ T * M without coordinate computations, which improves and generalizes the partial result obtained in Tomáš (2009) from m ≥ k to all cases of m. We also introduce the space J ( M, N) of A-jets and prove its rigidity in the sense of its coincidence with the classical jet space J ( M, N). [ABSTRACT FROM AUTHOR]

We first prove that a compact embedded minimal annulus in [ABSTRACT FROM AUTHOR]

We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to r-stability as well as to almost-Einstein hypersurfaces. [ABSTRACT FROM AUTHOR]

Given a Riemannian manifold with a weighted Poincaré inequality, in this paper, we will show some vanishing type theorems for p -harmonic ℓ -forms on such a manifold. We also prove a vanishing result on submanifolds in Euclidean space with flat normal bundle. Our results can be considered as generalizations of the work of Lam, Li–Wang, Lin, and Vieira (see Lam (2008), Li and Wang (2001), Lin (2015), Vieira (2016)). Moreover, we also prove a vanishing and splitting type theorem for p -harmonic functions on manifolds with Spin (9) holonomy provided a ( p , p , λ ) -Sobolev type inequality which can be considered as a general Poincaré inequality holds true. [ABSTRACT FROM AUTHOR]

Consider the supremal functional applied to $$W^{1,\infty }$$ maps $$u:\Omega \subseteq \mathbb {R}\longrightarrow \mathbb {R}^N$$ , $$N\ge 1$$ . Under certain assumptions on $$\mathscr {L}$$ , we prove for any given boundary data the existence of a map which is: Our method is based on $$L^p$$ approximations and stable a priori partial regularity estimates. For item ii) we utilise the recently proposed by the author notion of $$\mathcal {D}$$ -solutions in order to characterise the limit as a generalised solution. Our results are motivated from and apply to Data Assimilation in Meteorology. [ABSTRACT FROM AUTHOR]

The main analytical ingredients of the first part of this paper are two independent results: a theorem on approximation of W 2 , 2 solutions of the Monge–Ampère equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometric embeddings of 2d surface in R 3 . In the second part, we rigorously derive the Γ-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness h , where the depth of the shell scales like h α and the applied forces scale like h α + 2 , in the limit when h → 0 . We offer a full analysis of the problem in the parameter range α ∈ ( 1 / 2 , 1 ) . We also complete the analysis in some specific cases for the full range α ∈ ( 0 , 1 ) , applying the results of the first part of the paper. [ABSTRACT FROM AUTHOR]

The authors discuss the existence and uniqueness up to isometries of E of immersions ϕ: Ω ⊂ R → E with prescribed metric tensor field ( g ): Ω → S, and discuss the continuity of the mapping ( g ) → ϕ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping ( g ) → ϕ can be obtained by means of nonlinear Korn inequalities is shown. [ABSTRACT FROM AUTHOR]

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow. [ABSTRACT FROM AUTHOR]