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We prove that simply connected H-surfaces with small diameter in a 3-manifold necessarily concentrate at a critical point of the scalar curvature. Introduction Let (N, g) be a compact oriented Riemannian manifold. The aim of this article is to understand the behaviour of a sequence of surfaces ΣH ⊂ N with constant mean curvature H, refereed to as H-surfaces, when H → +∞. TheseH-surfaces naturally appear as boundaries of isoperimetric domains. Their existence is given by geometric measure theory [25], but we have no information about their topology or their location in the considered manifold except for some special manifolds like space forms where we have a classification of compact embedded H-surfaces (this is an extension of Aleksandrov theorem [1], see for instance [24]). It would be too ambitious for now to hope for a classification of these H-surfaces in a general compact manifold. However, in the particular case of minimal surfaces (H = 0), a rough classification can be obtained thanks to the works of Colding, Meeks, Minicozzi, Ros and Rosenberg and others. We will find an overview on this subject in the collective book edited by Hoffman [20] and the papers of Colding and Minicozzi [11], [12], [13] and [14]. This area of research is still very active motivated by its close links with the topology of 3-manifolds. In order to begin the description of the moduli space of H-surfaces, we look to the case of surfaces with small diameter (or large mean curvature). Up to perform dilation of the ambient space, we can normalize the mean curvature of these surfaces to be 1 and the ambient space becomes quasi-Euclidean. In this setting, an idea to obtain explicit examples of constant mean curvature surfaces is to pertub the constant mean curvature surfaces of the Euclidean space (i.e. round spheres but also connected sums of spheres and Delaunay surfaces) in order to get surfaces with constant mean curvature in our quasi-Euclidean space. This idea has been very successful and has led to many examples, see Ye [38], Butscher [6], Butscher-Mazzeo [7], Pacard [29] and Pacard-Xu [30]. But each of these constructions requires a condition on the geometry of the manifold at the point of concentration. A natural question then is the question of the necessity of this geometric condition. In fact if we were able to show that these conditions are necessary, we would have a clearer picture of the moduli space, at least for surfaces of small diameter. A first answer Paul Laurain UMPA-ENS Lyon 46 alle d’Italie 69364 Lyon Cedex 07. paul.laurain@umpa.ens-lyon.fr. 1

In the present paper we study multiplicity results for a Yamabe-type problem proposed by Escobar in 1992. We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, having null scalar curvature and constant mean curvature on the boundary. We use some standard results from the bifurcation theory to prove the existence of an infinite number of conformal classes with at least two non-homothetic Riemannian metrics of null scalar curvature and constant mean curvature on the boundary of the product manifold. In addition, we obtain a convergence result for bifurcating branches.

We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature. This adapts a method developed by Rugang Ye to construct foliations by surfaces of constant mean curvature., Minor modifications, to appear in Ann. Inst. Fourier (Grenoble)

In this paper, we consider the problem of the nonnegative scalar curvature (NNSC)-cobordism of Bartnik data $$\left( {\sum _1^{n - 1},{{\rm{\gamma }}_1},{H_1}} \right)$$ and $$\left( {\sum _2^{n - 1},{{\rm{\gamma }}_2},{H_2}} \right)$$ . We prove that given two metrics γ1 and γ2 on Sn−1 (3 ⩽ n ⩽ 7) with H1 fixed, then (Sn−1, γ1, H1) and (Sn−1, γ2, H2) admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough (see Theorem 1.3). Moreover, we show that for n = 3, a much weaker condition that the total mean curvature $$\int_{{S^2}}^{} {{H_2}d{\mu _{{\gamma _2}}}} $$ is large enough rules out NNSC-cobordisms (see Theorem 1.2); if we require the Gaussian curvature of γ2 to be positive, we get a criterion for nonexistence of the trivial NNSC-cobordism by using the Hawking mass and the Brown-York mass (see Theorem 1.1). For the general topology case, we prove that $$\left( {\Sigma _1^{n - 1},{{\rm{\gamma }}_1},0} \right)$$ and $$\left( {{\rm{\Sigma }}_2^{n - 1},{{\rm{\gamma }}_2},{H_2}} \right)$$ admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough (see Theorem 1.5).

In this paper, we establish a sharp integral inequality for n-dimensional closed spacelike submanifolds with constant scalar curvature immersed with parallel normalized mean curvature vector field in the de Sitter space $$\mathbb S_p^{n+p}$$ of index p, and we use it to characterize totally umbilical round spheres $$\mathbb S^n(r)$$ , with $$r>1$$ , of $$\mathbb S_1^{n+1}\hookrightarrow \mathbb S_p^{n+p}$$ . Our approach is based on a suitable lower estimate of the Cheng-Yau operator acting on the square norm of the traceless second fundamental form of such a spacelike submanifold.

We study hypersurfaces in the pseudo-Euclidean space, whose mean curvature vector satisfies the equation: Laplacian of the vector is parallel to the vector (with constant factor), and the second fundamental form has constant norm. We prove that every such hypersurface of diagonalizable shape operator with at most six distinct principal curvatures has constant mean curvature and constant scalar curvature, and if the above factor is zero then the hypersurface is minimal. We classify locally such non-minimal hypersurfaces with extremal value of the norm of the mean curvature vector. Further, we provide some examples of such hypersurfaces.

We study self-expanding solutions $M^m\subset\mathbb{R}^{n}$ of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function $|A|^2/|H|^2$ attains a local maximum, where $A$ denotes the second fundamental form and $H$ the mean curvature vector of $M$. If the pricipal normal $\xi=H/|H|$ is parallel in the normal bundle, then a similar result holds in higher codimension for the function $|A^\xi|^2/|H|^2$, where $A^\xi$ is the second fundamental form with respect to $\xi$. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension $2$ any mean convex self-expander that is asymptotic to a cone must be strictly convex., Comment: 2 figures

In this paper, the authors have established the necessary and sufficient conditions for the submanifolds of Kaehler product manifolds to be biharmonic. Moreover, the magnitude of scalar curvature for the hypersurfaces in a product of two unit spheres has been derived. Also, for the same product, the magnitude of the mean curvature vector for Lagrangian submanifolds has been estimated. Finally, the non-existence condition for totally complex Lagrangian submanifolds in a product of unit sphere and a hyperbolic space has been proved.

In this paper, we determine the sectional curvature K(X, Y) of a Legendre submanifold $$M^n$$ in Lorentzian Sasakian space forms $$\overline{M_1}^{2n+1}(k)$$ . Thus, we find the Ricci tensor $$\rho $$ and the scalar curvature $$\tau $$ of Legendre submanifold $$M^n$$ . From these, we get the equivalent condition with $$M^n$$ totally geodesic. Next, we prove that Legendre surfaces $$M^2$$ in Lorentzian Sasakian space forms $$\overline{M_1}^{5}(k)$$ with C-parallel mean curvature vector field are minimal or locally product of two curves. Moreover, we study Legendre surfaces whose mean curvature vector fields are eigenvectors of the Laplace operator (in normal bundle).

In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $$(\varSigma ,\gamma ,H)$$ . We prove that given a metric $$\gamma $$ on $${{\mathbf {S}}}^{n-1}$$ ( $$3\le n\le 7$$ ), $$({{\mathbf {S}}}^{n-1},\gamma ,H)$$ admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 4). Moreover, we prove that if $$\gamma $$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $${{\mathbf {S}}}^{n-1}$$ , then the much weaker condition that the total mean curvature $$\int _{{{\mathbf {S}}}^{n-1}}H\,{{\mathrm {d}}}\mu _\gamma $$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the $$\theta $$ -invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

Let $$M^n$$ be a complete n-dimensional Riemannian manifold and $$\Gamma _f$$ the graph of a $$C^2$$ -function f defined on a metric ball of $$M^n$$ . In the same spirit of the estimates obtained by Heinz for the mean and Gaussian curvatures of a surface in $${\mathbb {R}}^3$$ which is a graph over an open disk in the plane, we obtain in this work upper estimates for $$\inf |R|$$ , $$\inf |A|$$ and $$\inf |H_k|$$ , where R, |A| and $$H_k$$ are, respectively, the scalar curvature, the norm of the second fundamental form and the k-th mean curvature of $$\Gamma _f$$ . From our estimates we obtain several results for graphs over complete manifolds. For example, we prove that if $$M^n,\;n\ge 3,$$ is a complete noncompact Riemannian manifold with sectional curvature bounded below by a constant c, and $$\Gamma _f$$ is a graph over M with Ricci curvature less than c, then $$\inf |A|\le 3(n-2)\sqrt{-c}$$ . This result generalizes and improves a theorem of Chern for entire graphs in $$\mathbb R^{n+1}$$ .

In this paper we are interested in studying the set of positive solutions to the following Yamabe–type problem (⁎) { − Δ g u + a ( x ) u = 0 , in M , ∂ u ∂ η g + b ( x ) u = ( n − 2 ) u n n − 2 , on ∂ M , where ( M n , g ) is a compact Riemannian n−manifold, n ≥ 3 , with boundary, Δ g is the Laplace–Beltrami operator for g, η g is the pointing unit outward normal to ∂M, a ∈ C ∞ ( M ) and b ∈ C ∞ ( ∂ M ) . Assuming suitable hypotheses on the functions Ψ ( ξ ) = a ( ξ ) − n − 2 4 ( n − 1 ) R g ( ξ ) and Φ ( ξ ¯ ) = b ( ξ ¯ ) − n − 2 2 h g ( ξ ¯ ) , where R g denotes the scalar curvature of the metric g and h g is the mean curvature of ∂M, we prove some results of compactness for the set of positive solutions to the problem (⁎) .

We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a non-positively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture that a biharmonic submanifold in a sphere has constant mean curvature, and two more cases that support Chen's conjecture on biharmonic hypersurfaces (Corollaries 2.2,2.7)., 11 pages

Let Mn(n ≥ 4) be an oriented compact submanifold with parallel mean curvature in an (n + p)-dimensional complete simply connected Riemannian manifold Nn+p. Then there exists a constant δ(n, p) 2 (0, 1) such that if the sectional curvature of N satisfies $${\overline K _N} \in \;\,\left[ {\delta \left( {n,p} \right),\;1} \right]$$, and if M has a lower bound for Ricci curvature and an upper bound for scalar curvature, then N is isometric to Sn+p. Moreover, M is either a totally umbilic sphere $${S^n}({1 \over {\sqrt {1 + {H^2}} }})$$, a Clifford hypersurface $${S^m}({1 \over {\sqrt {2(1 + {H^2})} }})\; \times \;{S^m}({1 \over {\sqrt {2(1 + {H^2})} }})$$ in the totally umbilic sphere $${S^{n+1}}({1 \over {\sqrt {1 + {H^2}} }})$$ with n = 2m, or $${\mathbb{C}{\rm{P}}^2}\left( {{4 \over 3}\left( {1 + {H^2}} \right)} \right)$$ in $${S^7}({1 \over {\sqrt {1 + {H^2}} }})$$. This is a generalization of Ejiri’s rigidity theorem.

We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on marginally outer trapped surfaces applied to appropriate initial data sets. One of the results involves an analysis of Obata's equation on manifolds with boundary. This result is relevant to recent work of Lan-Hsuan Huang and the second author concerning the rigidity of asymptotically locally hyperbolic manifolds with zero mass., 15 pages; v2: Introduction revised, other small changes; results unchanged. To appear in Proc. Amer. Math. Soc

Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the Bartnik mass of such surfaces, without assumptions on the integral of the squared mean curvature. If the ambient manifold has negative scalar curvature, our method also applies and yields estimates on the hyperbolic Bartnik mass of these surfaces., Comment: version accepted by Math. Res. Lett

In order to derive the best possible estimates of the total mean curvature of a compact submanifold in a Euclidean space in terms of spectral geometry, in the late 1970s, Bang-Yen Chen introduced the theory of finite-type submanifolds, which could be viewed as $$\lambda $$ -biharmonic submanifolds in the sense of $$\lambda $$ -biharmonic maps. Interestingly, Chen proposed in 1991 the following problem (Chen in Soochow J Math 17:2:169–188, 1991, Problem 12): “Determine all submanifolds of Euclidean spaces which are of null 2-type. In particular, classify null 2-type hypersurfaces in Euclidean spaces.” In this paper, we give further support evidence to the above problem. We are able to prove that a linear Weingarten null 2-type or $$\lambda $$ -biharmonic hypersurface $$M^n$$ of a Euclidean space $$\mathbb R^{n+1}$$ has constant mean curvature and constant scalar curvature provided $$n

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.

Given a compact manifold with boundary M of dimension m ≥ 3 and a nondegenerate Riemannian metric g ∗ having null scalar curvature, constant mean curvature, and unitary volume on the boundary, we show that the set of Riemannian metrics with null scalar curvature, constant mean curvature, and unitary volume on the boundary, near to g ∗ is an embedded submanifold of the manifold of all Riemannian metrics on M . Additionally, such submanifold is strongly transversal to the conformal classes. We also prove, using recent results of compactness, that conformal classes of metrics closed to g ∗ contain only one of these metrics.

We study biharmonic hypersurfaces in the space forms $$\overline{M}^{6}(c)$$ with at most four distinct principal curvatures and whose second fundamental form is of constant norm. We prove that every such biharmonic hypersurface in $$\overline{M}^{6}(c)$$ has constant mean curvature and constant scalar curvature. In particular, every such biharmonic hypersurface in $$\mathbb {S}^{6}(1)$$ has constant mean curvature and constant scalar curvature. Every such biharmonic hypersurface in Euclidean space $$E^6$$ and in hyperbolic space $$\mathbb {H}^{6}$$ must be minimal and have constant scalar curvature.

We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $$\Sigma $$ with non-positive Yamabe invariant in a Riemannian n-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that $$\Sigma $$ is locally volume-minimizing in a manifold $$M^n$$ with scalar curvature bounded from below by a non-positive constant and mean convex boundary, we conclude that locally M splits along $$\Sigma $$ . In the case that the scalar curvature of M is at least $$-n(n-1)$$ and $$\Sigma $$ locally minimizes a certain functional inspired by works of Yau [35] and Andersson-Galloway [4], a neighbourhood of $$\Sigma $$ in M is isometric to $$((-\varepsilon , \varepsilon ) \times \Sigma ,\mathrm{{d}}t^{2}+e^{2t}g)$$ , where g is Ricci flat with totally geodesic boundary.

We show that under suitable non-degeneracy conditions, complete gradient flow lines of the scalar curvature functional of a riemannian manifold perturb into eternal forced mean curvature flows with large forcing term.

In this paper we prove classification results to elliptic fully nonlinear conformal equations in certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball in S n ) of dimension n ≥ 2 with prescribed constant mean curvature on its boundary, and annular domains with minimal boundary. Our results extend previous ones concerning more general curvature functions on the eigenvalues of the Schouten tensor than scalar curvature.