Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. The Osserman Conjecture asserts that any Osserman manifold is either flat or rank-one symmetric. We prove that both the Osserman Conjecture and its conformal version, the Conformal Osserman Conjecture, are true, modulo a certain assumption on algebraic curvature tensors in ${\mathbb {R}^{16}}$ . As a consequence, we show that a Riemannian manifold having the same Weyl tensor as a rank-one symmetric space is conformally equivalent to it.     

On projective N-curvature inheritance

The concept of a Lie recurrence was introduced by the first author?[6]. It is an infinitesimal transformation $\overline{x}^{i}={x}^{i}+\varepsilon {v}^{i}({x}^{j})$ with respect to which the Lie derivative of a curvature tensor is proportional to itself. Apart from other results related to a Lie recurrence, it was established that the Weyl projective curvature tensor is Lie recurrent with respect to a Lie recurrence but its converse is not necessarily true. However, an infinitesimal transformation with respect to which the Weyl projective curvature tensor and the Ricci tensor are Lie recurrent, is necessarily a Lie recurrence. Singh?[12] studied an infinitesimal transformation with respect to which the Lie derivative of the curvature tensor is proportional to itself and called such transformation as curvature inheritance. Obviously, a curvature inheritance is nothing but a Lie recurrence. Singh?[13] also considered a curvature inheritance which is a projective motion and called it a projective curvature inheritance. Gatoto and Singh [1,2] studied $\widetilde{K}$ -curvature inheritance and projective $\widetilde{K}$ -curvature inheritance. Pandey and Pandey?[9] studied $\widetilde{K}$ projective Lie recurrence. Mishra and Yadav?[3] studied projective curvature inheritance in an NP-F _n . In the present paper we have established that an infinitesimal transformation in a Finsler space is Lie recurrence if and only if the normal projective curvature tensor is Lie recurrent. A part from this result we have generalized almost all theorems of Mishra and Yadav?[3].     

Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature

In this paper, we show that, for every biharmonic submanifold (M, g) of a Riemannian manifold (N, h) with non-positive sectional curvature, if ${\int_M\vert \eta \vert^2 v_g < \infty}$ , then (M, g) is minimal in (N, h), i.e., ${\eta\equiv0}$ , where η is the mean curvature tensor field of (M, g) in (N, h). This result gives an affirmative answer under the condition ${\int_M\vert \eta \vert^2 v_g < \infty}$ to the following generalized Chen’s conjecture: every biharmonic submanifold of a Riemannian manifold with non-positive sectional curvature must be minimal. The conjecture turned out false in case of an incomplete Riemannian manifold (M, g) by a counter example of Ou and Tang (in The generalized Chen’s conjecture on biharmonic sub-manifolds is false, a preprint, 2010).     

ε_0-Regularity for Mean Curvature Flow from Surface to Flat Riemannian Manifold

In this paper we prove an ε0-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫Σ0 |A|2 ≤ε0 and the initial area μ0(Σ0) is not large, then along the mean curvature flow, we have ∫Σt|A|2 ≤ε0. As an application, we obtain the long time existence and convergence result of the mean curvature flow.     

Some projectively flat (α, β)-metrics

In this paper, we study a class of Finsler metrics in the form $F = \alpha + \varepsilon \beta + 2k\tfrac{{\beta ^2 }}{\alpha } - \tfrac{{k^2 \beta ^4 }}{{3\alpha ^3 }}$ , where $\alpha = \sqrt {\alpha _{ij} y^i y^j } $ is a Riemannian metric, β = b _i y ⁱ is a 1-form, and ε and k ≠ 0 are constants. We obtain a sufficient and necessary condition for F to be locally projectively flat and give the non-trivial special solutions. Moreover, it is proved that such projectively flat Finsler metrics with the constant flag curvature must be locally Minkowskian.     

Liouville-Type Theorems and Applications to Geometry on Complete Riemannian Manifolds

On a complete Riemannian manifold M with Ricci curvature satisfying $$\mathrm{Ric}(\nabla r,\nabla r) \geq -Ar^2(\log r)^2(\log(\log r))^2\cdots (\log^{k}r)^2$$ for r?1, where A>0 is a constant, and r is the distance from an arbitrarily fixed point in M, we prove some Liouville-type theorems for a C ² function f:M→? satisfying Δf≥F(f) for a function F:?→?. As an application, we obtain a C ⁰ estimate of a spinor satisfying the Seiberg–Witten equations on such a manifold of dimension 4. We also give applications to the conformal transformation of the scalar curvature and isometric immersions of such a manifold.     

On the stability of the L p -norm of the Riemannian curvature tensor

We consider the Riemannian functional \(\mathcal {R}_{p}(g)={\int }_{M}|R(g)|^{p}dv_{g}\) defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M where R(g) and dv _g denote the corresponding Riemannian curvature tensor and volume form and p ∈ (0, ∞). First we prove that the Riemannian metrics with non-zero constant sectional curvature are strictly stable for \(\mathcal {R}_{p}\) for certain values of p. Then we conclude that they are strict local minimizers for \(\mathcal {R}_{p}\) for those values of p. Finally generalizing this result we prove that product of space forms of same type and dimension are strict local minimizer for \(\mathcal {R}_{p}\) for certain values of p.     

Geometric rigidity for analytic estimates of Müller–?verák

In the paper Müller–?verák (J Differ Geom 42(2):229–258, 1995) conformally immersed surfaces with finite total curvature were studied. In particular it was shown that surfaces with total curvature ${\int_{\Sigma} |A|^2 < 8 \pi}$ in dimension three were embedded and conformal to the plane with one end. Here, using techniques from Kuwert–Li (W ^2,2-conformal immersions of a closed Riemann surface into R ⁿ . arXiv:1007.3967v2 [math.DG], 2010), we will show that if the total curvature ${ \int_{\Sigma}|A|^2\leq8\pi}$ , then we are either embedded and conformal to the plane, isometric to a catenoid or isometric to Enneper’s minimal surface. In fact the technique of our proof shows that if we are conformal to the plane, then if n?≥ 3 and ${ \int_{\Sigma} | A|^{2}\leq 16 \pi }$ then Σ is embedded or Σ is the image of a generalized catenoid inverted at a point on the catenoid. In order to prove these theorems, we prove a Gauss–Bonnet theorem for surfaces with complete ends and isolated finite area singularities which extends a theorem of Jorge-Meeks (Topology 22(2):203–221, 1983). Using this theorem, we then prove an inversion formula for the Willmore energy.     

On Einstein Matsumoto metrics

We study a special class of Finsler metrics,namely,Matsumoto metrics F=α2α-β,whereαis a Riemannian metric andβis a 1-form on a manifold M.We prove that F is a(weak)Einstein metric if and only ifαis Ricci flat andβis a parallel 1-form with respect toα.In this case,F is Ricci flat and Berwaldian.As an application,we determine the local structure and prove the 3-dimensional rigidity theorem for a(weak)Einstein Matsumoto metric.     

Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles S _r M of a Riemannian manifold M, g of \({{\rm dim}\geq3}\) , assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure I ^G and symplectic structure \({\omega^G}\) on the manifold TM, generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel–Whitney characteristic classes of the manifolds TM and S _r M.     

Caccioppoli’s inequalities on constant mean curvature hypersurfaces in Riemannian manifolds

We prove some Caccioppoli’s inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant mean curvature hypersurface of a Riemannian manifold, satisfying some curvature conditions. This allows us to unify and clarify many results scattered in the literature and to obtain some new results. For example, we prove that there is no stable, complete, noncompact hypersurface in ${{\mathbb R}^{n+1}, n \leq 5}$ , with constant mean curvature ${H \not=0}$ , provided that, for suitable p, the L ^p norm of the traceless part of second fundamental form satisfies some growth condition.     

An inequality between Willmore functional and Weyl functional for submanifolds in space forms

Let \({\phi : M \to R^{n+p}(c)}\) be an n-dimensional submanifold in an (n + p)-dimensional space form R ^n+p(c) with the induced metric g. Willmore functional of \({\phi}\) is \({W(\phi) = \int_{M}(S - nH^{2})^{n/2}dv}\) , where \({S = \sum_{\alpha,i, j}(h^{\alpha}_{ij} )^2}\) is the square of the length of the second fundamental form, H is the mean curvature of M. The Weyl functional of (M, g) is \({\nu(g) = \int_{M}|W_{g}|^{n/2}dv}\) , where \({|W_{g}|^{2} = \sum_{i, j,k,l} W^{2}_{ijkl}}\) and W _ijkl are the components of the Weyl curvature tensor W _g of (M, g). In this paper, we discover an inequality relation between Willmore functional \({W(\phi)}\) and Weyl funtional ν(g).     

On the regularity of solutions of model nonlinear elliptic systems with the oblique derivative type boundary condition

Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $B_1^ + = B_1 (0) \cap \{ x_n > 0\} \subset $ ? ⁿ , with the oblique derivative type boundary condition on $\Gamma _1 = B_1 (0) \cap \{ x_n = 0\} $ . For solutionsu∈H ¹(B ₁ ⁺ ) of systems of the form $\frac{d}{{dx_\alpha }}a_\alpha ^k (u_x ) = 0, k \leqslant {\rm N}$ , it is proved that the derivatives u_x are Hölder in $B_1^ + \cup \Gamma _1 )\backslash \Sigma $ , where H_n?p(σ)=0,p>2. It is shown for continuous solutions u from H¹(B₁/⁺) of systems $\frac{d}{{dx_\alpha }}a_\alpha ^k (u,u_x ) = 0$ that the derivatives u_x are Hölder on the set $(B_1^ + \cup \Gamma _1 )\backslash \Sigma , dim_\kappa \Sigma \leqslant n - 2$ . Bibliography: 13 titles.     

Symplectic mean curvature flow in CP 2

Let Σ be an immersed symplectic surface in CP ² with constant holomorphic sectional curvature k > 0. Suppose Σ evolves along the mean curvature flow in CP ². In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies ${|A|^2 \leq \lambda|H|^2 + \frac{2\lambda-1}{\lambda}k}$ and ${\cos\alpha\geq\sqrt{\frac{7\lambda-3}{3\lambda}}\left(\frac{1}{2} < \lambda\leq\frac{2}{3}\right) {\rm or} |A|^2\leq \frac{2}{3}|H|^2+\frac{4}{5}k\cos\alpha\, {\rm and} \cos\alpha\geq 1-\varepsilon}$ , for some ${\varepsilon}$ .     

Levi Harmonic Maps of Contact Riemannian Manifolds

We study Levi harmonic maps, i.e., C ^∞ solutions f:M→M′ to \(\tau_{\mathcal{H}} (f) \equiv \operatorname{trace}_{g} ( \varPi_{\mathcal{H}}\beta_{f} ) = 0\) , where (M,η,g) is an (almost) contact (semi) Riemannian manifold, M′ is a (semi) Riemannian manifold, β _f is the second fundamental form of f, and \(\varPi_{\mathcal{H}} \beta_{f}\) is the restriction of β _f to the Levi distribution \({\mathcal{H}} = \operatorname{Ker}(\eta)\) . Many examples are exhibited, e.g., the Hopf vector field on the unit sphere S ²ⁿ⁺¹, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannian manifold of constant curvature 1 are Levi harmonic maps. A CR map f of contact (semi) Riemannian manifolds (with spacelike Reeb fields) is pseudoharmonic if and only if f is Levi harmonic. We give a variational interpretation of Levi harmonicity. Any Levi harmonic morphism is shown to be a Levi harmonic map.     

Existence of solutions for the critical elliptic system with inverse square potentials

Let Ω ? 0 be an open bounded domain in R ^N (N ≥ 3) and $2^* (s) = \tfrac{{2(N - s)}} {{N - 2}}$ , 0 < s < 2. We consider the following elliptic system of two equations in H ₀ ¹ (Ω) × H ₀ ¹ (Ω): $$- \Delta u - t\frac{u} {{\left| x \right|^2 }} = \frac{{2\alpha }} {{\alpha + \beta }}\frac{{\left| u \right|^{\alpha - 2} u\left| v \right|^\beta }} {{\left| x \right|^s }} + \lambda u, - \Delta v - t\frac{v} {{\left| x \right|^2 }} = \frac{{2\beta }} {{\alpha + \beta }}\frac{{\left| u \right|^\alpha \left| v \right|^{\beta - 2} v}} {{\left| x \right|^s }} + \mu v,$$ where λ, µ > 0 and α, β > 1 satisfy α + β = 2*(s). Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.     

Arc curvature in metric spaces

Concepts for curvature of arcs in metric geometry (specifically, Menger curvature κ _M , Haantjes-Finsler curvature κ _H , and transverse curvature κ _T introduced earlier by the author) are compared with respect to existence and numerical values. If a metric space satisfies a certain metric inequality shared in particular by Riemannian spaces, then the pointwise existence of κ _M on any arc implies that of κ _T and the two are equal. In a Minkowskian plane X with strictly convex unit sphere whose boundary U has a C ² polar representation ρ=ρ(θ), and with \(\bar \kappa _M\) and \(\bar \kappa _M\) the Menger and transverse curvatures relative to the underlying Euclidean metric, the following formulas are proved: At any point p on an arc at which \(\bar \kappa _M\) and \(\bar \kappa _M\) exist, $$\kappa _M = \bar \kappa _M \sqrt {\rho ^{2 + } 2\rho ^{'2 - } \rho \rho }$$ and $$\kappa _T = \bar \kappa _T \frac{{\sigma _1^{3/2} (T_p )}}{{\sigma _2 (T_p ,T_p^ \bot )}},$$ where T _pis the tangent at p, T ^⊥ _pthat line to which T _pis metrically perpendicular, and σ₁ and σ₂ are certain real-valued functions defined on lines of X. The result of this is that if κ* is the classical curvature of U _p≡U+p at U _p∩ T _p, $$\frac{{\kappa _M^2 }}{{\kappa _T^2 }} = \frac{{\kappa ^ * \sigma _1^{3/2} (T_p^ \bot )}}{{\sigma _2 \left( {T_p ,T_p^ \bot } \right)}},$$ from which it follows that the values of κ _M and κ _T are not equal for metric spaces in general even when both exist.     

Concordance and Isotopy of Metrics with Positive Scalar Curvature

Two positive scalar curvature metrics g ₀, g ₁ on a manifold M are psc-isotopic if they are homotopic through metrics of positive scalar curvature. It is well known that if two metrics g ₀, g ₁ of positive scalar curvature on a closed compact manifold M are psc-isotopic, then they are psc-concordant: i.e., there exists a metric ${\bar{g}}$ of positive scalar curvature on the cylinder ${M \times I}$ which extends the metrics g ₀ on ${M \times \{0\}}$ and g ₁ on ${M \times \{1\}}$ and is a product metric near the boundary. The main result of the paper is that if psc-metrics g ₀, g ₁ on M are psc-concordant, then there exists a diffeomorphism ${\Phi : M \times I \rightarrow M \times I}$ with ${\Phi|_{M \times \{0\}} = Id}$ (a pseudo-isotopy) such that the metrics g ₀ and ${(\Phi|_{M \times \{1\}})^{*}g_{1}}$ are psc-isotopic. In particular, for a simply connected manifold M with dim M ≥  5, psc-metrics g ₀, g ₁ are psc-isotopic if and only if they are psc-concordant. To prove these results, we employ a combination of relevant methods: surgery tools related to the Gromov–Lawson construction, classic results on isotopy and pseudo-isotopy of diffeomorphisms, standard geometric analysis related to the conformal Laplacian, and the Ricci flow.     

Eigenvalue problems for quasilinear elliptic systems with limiting nonlinearity

In this paper, we consider a class of quasilinear elliptic eigenvalue problems with limiting nonlinearity. First, we use the concentration-compactness principle to get the existence of a minimum uεH ₀ ¹ (ω,R ^N ) of the minimization problem \(I_{\lambda _0 } = \inf \{ \smallint _\Omega (a_{\alpha \beta } (x)g_{ij} (u)D_\alpha u^i D_\beta u^j + h(x)|u|^2 )|u \in H_0^1 (\Omega ,R^N ),\smallint _\Omega |u|^{2n/(n - 2)} = \lambda _0 \} ;\) then we apply the reverse Hölder inequality to prove thatuεL ^∞ (ω, R ^N ).     

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Y ⁿ denote the Gromov-Hausdorff limit $M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}$ of v-noncollapsed Riemannian manifolds with ${\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)$ . The singular set $\mathcal {S}\subset Y$ has a stratification $\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}$ , where $y\in \mathcal {S}^{k}$ if no tangent cone at y splits off a factor ? ^k+1 isometrically. Here, we define for all η>0, 0<r≤1, the k-th effective singular stratum $\mathcal {S}^{k}_{\eta,r}$ satisfying $\bigcup_{\eta}\bigcap_{r} \,\mathcal {S}^{k}_{\eta,r}= \mathcal {S}^{k}$ . Sharpening the known Hausdorff dimension bound $\dim\, \mathcal {S}^{k}\leq k$ , we prove that for all y, the volume of the r-tubular neighborhood of $\mathcal {S}^{k}_{\eta,r}$ satisfies ${\mathrm {Vol}}(T_{r}(\mathcal {S}^{k}_{\eta,r})\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},\eta)r^{n-k-\eta}$ . The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let $\mathcal {B}_{r}$ denote the set of points at which the C ²-harmonic radius is ≤r. If also the $M^{n}_{i}$ are Kähler-Einstein with L ₂ curvature bound, $\| Rm\|_{L_{2}}\leq C$ , then ${\mathrm {Vol}}( \mathcal {B}_{r}\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},C)r^{4}$ for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the $M^{n}_{i}$ , we obtain a slightly weaker volume bound on $\mathcal {B}_{r}$ which yields an a priori L _p curvature bound for all p<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.