Search

One important problem in Finsler geometry is that of classifying Finsler metrics of scalar curvature. By investigating the second-order differential equation for a class of Randers metrics with isotropic S -curvature, we give a global classification of these metrics of scalar curvature, generalizing a theorem previously only known in the case of locally projectively flat Randers metrics.

It is well known that critical points of the total scalar curvature functional S on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of S is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is isometric to a standard sphere. In this paper we investigate the relationship between the first Betti number and stable minimal surfaces, and study the analytic properties of stable minimal surfaces in M for n = 3.

We consider a C1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed p-mean curvature \({H\in C^{0},}\) we show that any characteristic curve is C2 smooth and its (line) curvature equals − H in the nonsingular domain. By introducing characteristic coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are C2 smooth. We also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. These results can be applied to more general situations.

In the thesis, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kahler metric. In fact, this extremal Kahler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence classes of the blow-up of P 2 at one or two points. The explicit construction of compact complex surfaces with constant scalar curvature Kahler metrics in different deformation equivalence classes is given. The main tool repeatedly applied here is the gluing theorem of C. Arezzo and F. Pacard which states that the blow-up/resolution of a compact manifold/orbifold of discrete type, which admits cscK metrics, still admits cscK metrics.

Explicit formulas for the G 2 -components of the Riemannian curvature tensor on a manifold with a G 2 -structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact G 2 -manifold with closed fundamental form forces the three-form to be parallel. A topological obstruction for the existence of a G 2 -structure with closed fundamental form is obtained in terms of the integral norms of the curvature components. We produce integral inequalities for closed G 2 -manifold and investigate limiting cases. We make a study of warped products and cohomogeneity-one G 2 -manifolds. As a consequence every Fernandez–Gray type of G 2 -structure whose scalar curvature vanishes may be realized such that the metric has holonomy contained in G 2 .

We construct a natural L 2 -metric on the perturbed Seiberg–Witten moduli spaces M μ + of a compact 4-manifold M , and we study the resulting Riemannian geometry of M μ + . We derive a formula which expresses the sectional curvature of M μ + in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case M is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U ( 1 ) bundle P → M μ + such that the bundle projection becomes a Riemannian submersion. On a Kahler surface M , the L 2 -metric on M μ + coincides with the natural Kahler metric on moduli spaces of vortices.

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature $K$. If $s$ is the scalar curvature and $W_+$ is the self-dual part of Weyl tensor, then it will be shown that there is no metric $g$ on $S^2 \times S^2$ with both (i) $K > 0$ and (ii) $ {1/6} s - W_+ \ge 0$. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: ``If a simply-connected, closed 4-manifold $M^4$ admits a metric $g$ of non-negative curvature operator, then $M^4$ is one of $S^4$, $\Bbb CP^2$ and $S^2 \times S^2$". Our method is different from Hamilton's and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.

We show that a Riemannian metric on a 4-dimensional smooth manifold is of constant scalar curvature if and only if it is a critical metric of the restricted Schouten functional.

We consider the problem of constructing scalar particle wave equations in Riemannian spaces with external gauge fields whose symmetry group is the group of motions of the Riemannian space.

Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume functional. In this paper, we examine the critical points of the total 2k-th Gauss–Bonnet curvature functional, called 2k-minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the (2k + 1)-mean curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to 2k-minimal submanifolds.

Our main result in this paper establishes that if G is a compact Lie subgroup of the isometry group of a compact Riemannian manifold M acting with cohomogeneity one in M and either G has no singular orbits or the singular orbits of G have dimension at most n−3, then the unit vector field N orthogonal to the principal orbits of G is weakly smooth and is a critical point of the energy functional acting on the unit normal vector fields of M. A formula for the energy of N in terms of the of integral of the Ricci curvature of M and of the integral of the square of the mean curvature of the principal orbits of G is obtained as well. In the case that M is the sphere and G the orthogonal group it is known that that N is minimizer. It is an open question if N is a minimizer in general.

On a Riemannian manifold of nonpositive sectional curvature (Cartan-Hadamard-type manifold), we consider a parabolic equation. The second boundary-value problem for this equation is set in a bounded domain whose surface is a smooth submanifold. We prove that the gradient of the simple-layer potential for this problem has a jump when passing across the submanifold, similarly to its behavior in a Euclidean space.

By variational methods, for a kind of Yamabe problem whose scalar curvature vanishes in the unit ball B N and on the boundary S N-1 the mean curvature is prescribed, we construct multi-peak solutions whose maxima are located on the boundary as the parameter tends to 0+ under certain assumptions. We also obtain the asymptotic behaviors of the solutions.

We show that every symplectic manifold of dimension ≥ 4 admits a complete compatible almost Kahler metric of negative scalar curvature. And we discuss the C0-closure of the set of almost Kahler metrics of negative scalar curvature. Some local versions are also proved.

The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub-and supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub-and supersolutions are established.

Suppose that M is a compact orientable hypersurface embedded in a compact n-dimensional orientable Riemannian manifold N. Suppose that the Ricci curvature of N is bounded below by a positive constant k. We show that 2 λ 1 > k − ( n − 1 ) max M | H | where λ 1 is the first eigenvalue of the Laplacian of M and H is the mean curvature of M.

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that Ric M ⩾ − 4 ( p − 1 ) p 2 μ 0 at all x ∈ M and > − 4 ( p − 1 ) p 2 μ 0 at some point x 0 ∈ M , where μ 0 > 0 is the least eigenvalue of the Laplacian acting on L 2 -functions on M. Let 2 ⩽ q ⩽ p . Then any q-harmonic map ϕ : M → N of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism ϕ : M → N of finite q-energy is constant.

Consider an n n -dimensional smooth Riemannian manifold ( M n , g ) (M^n,g) with a given smooth measure m m on it. We call such a triple ( M n , g , m ) (M^n,g,m) a Riemannian measure space. Perelman introduced a variant of scalar curvature in his recent work on solving Poincaré’s conjecture P ( g ) = R ∞ m ( g ) = R ( g ) − 2 Δ g l o g ϕ − | ∇ l o g ϕ | g 2 P(g)=R^m_\infty (g) = R(g) - 2\Delta _g log\phi - |\nabla log\phi |^2_g , where d m = ϕ d v o l ( g ) dm = \phi dvol(g) and R R is the scalar curvature of ( M n , g ) (M^n,g) . In this note, we study the topological obstruction for the ϕ \phi -stable minimal submanifold with positive P P -scalar curvature in dimension three under the setting of manifolds with density.

Let M M be an n n -dimensional compact hypersurface with constant scalar curvature n ( n − 1 ) r n(n-1)r , r > 1 r> 1 , in a unit sphere S n + 1 ( 1 ) S^{n+1}(1) . We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral ∫ M H d M \int _MHdM of the mean curvature H H . In this paper, we first study the eigenvalue of the Jacobi operator J s J_s of M M . We derive an optimal upper bound for the first eigenvalue of J s J_s , and this bound is attained if and only if M M is a totally umbilical and non-totally geodesic hypersurface or M M is a Riemannian product S m ( c ) × S n − m ( 1 − c 2 ) S^m(c)\times S^{n-m}(\sqrt {1-c^2}) , 1 ≤ m ≤ n − 1 1\leq m\leq n-1 .

The energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

For a compact Riemannian manifold M , we obtain an explicit upper bound of the volume entropy with an integral of Ricci curvature on M and a volume ratio between two balls in the universal covering space.

In this article a nonsingular asymptotic distribution is derived for a broad class of underlying distributions on a Riemannian manifold in relation to its curvature. Also, the asymptotic dispersion is explicitly related to curvature. These results are applied and further strengthened for the planar shape space of k-ads.

In this paper, we introduce a 6-dimensional example of non-compact complete Einstein non-Kahler almost-Kahler manifold G with negative scalar curvature which was constructed by Apostolov-Draghici- Moroianu ([5]) and discuss the geometric structures.

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0., 4 pages

In this paper we construct constant scalar curvature metrics on the generalized connected sum M = M 1 ♯ K M 2 of two compact Riemannian manifolds (M 1,g 1) and (M 2,g 2) along a common Riemannian submanifold (K,g K ), in the case where the codimension of K is ≥ 3 and the manifolds M 1 and M 2 carry the same nonzero constant scalar curvature S. This yields a generalization of the Joyce's results for point-wise connected sums.

We describe some recent results concerning the inverse curvature problem, that is, the existence and description of metrics with prescribed curvature, focusing on the low-dimensional homogeneous cases.

This article mainly contains two theorems. In the first theorem, we obtain an upper bound for the area of isoperimetric surface that encloses small volume in a compact 3-manifold of which the scalar curvature is not less than the scalar curvature of corresponding space form. We also indicate that the 3-manifold must have constant sectional curvature when the upper bound is attained. In the second theorem, we assume the closed 3-manifold has nonnegative Ricci curvature, finite fundamental group and its scalar curvature is not less than $6$. Under these conditions, if the area of a cetain isoperimetric surface lies in a specific interval, then we will have an upper bound estimate for the volume of the 3-manifold.

In this paper, we study Finsler metrics of scalar flag curvature. We find that a non-Riemannian quantity is closely related to the flag curvature. We show that the flag curvature is weakly isotropic if and only if this non-Riemannian quantity takes a special form. This will lead to a better understanding on Finsler metrics of scalar flag curvature.

In this paper, we consider the problem of achieving constant scalar curvature on warped product manifolds according to fiber manifolds with constant scalar curvature.

We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold., final version

We discuss existence and multiplicity of solutions of the one-dimensional autonomous prescribed curvature problem , u(0) = 0, u(1) = 0, depending on the behaviour at the origin and at infinity of the function f. We consider solutions that are possibly discontinuous at the points where they attain the value zero.

We study the geometry of orthonormal frame bundles OM over Riemannian manifolds (M, g). The former are equipped with some modifications $$\tilde g_c$$ of the Sasaki-Mok metric $$\tilde g$$ depending on one real parameter c ≠ 0. The metrics $$\tilde g_c$$ are “strongly invariant” in some special sense. In particular, we consider the case when (M, g) is a space of constant sectional curvature K. Then, for dim M > 2, we find always, among the metrics $$\tilde g_c$$ , two strongly invariant Einstein metrics on OM which are Riemannian for K > 0 and pseudo-Riemannian for K

We prove firstly the classification theorem for p-harmonic morphisms between Euclidean domains. Secondly, we show that if \(\phi: M\to N\) is a p-harmonic morphism (p ≥ 2) from a complete Riemannian manifold M of nonnegative Ricci curvature into a Riemannian manifold N of non-positive scalar curvature such that the Lq-energy is finite, then \(\phi\) is constant, which improve the corresponding result due to G. Choi, G. Yun in (Geometriae Dedicata 101 (2003), 53–59).

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds ( M , g ) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.

We show that if the Ricci curvature of a left-invariant metric g on S3 is greater than that of the standard metric g0, then the eigenvalues of Δg are greater than the corresponding eigenvalues of Δgo.

Let M be an m dimensional submanifold in the Euclidean space Rn and H be the mean curvature ofM. We obtain some low geometric estimates of the total squaremean curvature ∫M H2dσ. The low bounds are geometric invariants involving the volume of M, the total scalar curvature of M, the Euler characteristic and the circumscribed ball of M.

The aim of the paper is to show how to explicitly express the function of sectional curvature with the first and second derivatives of the problem's functions in the case of submanifolds determined by equality constraints in the n-dimensional Euclidean space endowed with the induced Riemannian metric, which is followed by the formulation of the minimization problem of sectional curvature at an arbitrary point of the given submanifold as a global minimization one on a Stiefel manifold. Based on the results, the sectional curvatures of Stiefel manifolds are analysed and the maximal and minimal sectional curvatures on an ellipsoid are determined.

In this paper, we consider a boundary value problem associated with the conformal deformation of metrics on the unit ball B n , n ≥ 3 . Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove the existence of a metric, conformally equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary.

We give existence results for solutions of the prescribed scalar curvature equation on S 3 , when the curvature function is a positive Morse function and satisfies an index-count condition.

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of nonpositive scalar curvature. Let μ0 be the least eigenvalue of the Laplacian acting on L2-functions on M . We show that if RicM ≥ −μ0 at all x ∈ M and either RicM > −μ0 at some point x0 or Vol(M) is infinite, then every harmonic morphism φ : M → N of finite energy is constant.

In this paper we study non-simply connected Riemannian manifolds of constant positive curvature which have an orbit of codimension one under the action of a connected closed Lie subgroup of isometries. When the action is reducible we characterize the orbits explicitly. We also prove that in some cases the manifold is homogeneous.

The goal of this paper is to extend to two-dimensional optimal control systems with scalar input the classical notion of Gaussian curvature of two-dimensional Riemannian surface using the Cartan's moving frame method. This notion was already introduced by A. A. Agrachev and R. V. Gamkrelidze for more general control systems using a purely variational approach. Then we will see that the ``control'' analogue to Gaussian curvature reflects similar intrinsic properties of the extremal flow. In particular if the curvature is negative, arbitrarily long segment of extremals are locally optimal. Finally, we will define and characterize flat control systems., Comment: 7 pages

A two-parameter family of invariant almost-complex structures J α,c is given on the homogeneous space M × M’ = U(n + 1)/U(n) × U(p + 1)/U(p); all these structures are integrable. We consider all invariant Riemannian metrics on the homogeneous space M × M’. They depend on five parameters and are Hermitian with respect to some complex structure J α,c . In this paper, we calculate the Ricci tensor, scalar curvature, and obtain estimates of the sectional curvature for any metric on M × M’. All the invariant metrics of nonnegative curvature are described. We obtain the extremal values of the scalar curvature functional on the four-parameter family of metrics g α,c,λ,λ’;1 .

We show that any closed symplectic four-dimensional manifold (M, ω) admits an almost Kahler metric of negative scalar curvature compatible with ω.

Let (M,g) be a compact Riemannian manifold of dimension n \geq 3. The Compactness Conjecture asserts that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M,g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dimensions n \geq 52., Published paper

In this paper, we estimate the supremum of Perelman’s λ-functional λM(g) on Riemannian 4-manifold (M, g) by using the Seiberg-Witten equations. Among other things, we prove that, for a compact Kahler-Einstein complex surface (M, J, g0) with negative scalar curvature, (i) if g1 is a Riemannian metric on M with λM(g1) = λM(g0), then \(Vol_{g_1 } \) (M) ⩾ \(Vol_{g_0 } \) (M). Moreover, the equality holds if and only if g1 is also a Kahler-Einstein metric with negative scalar curvature. (ii) If {gt}, t ∈ [−1, 1], is a family of Einstein metrics on M with initial metric g0, then gt is a Kahler-Einstein metric with negative scalar curvature.

We study the structure of the minimum set of the normal curvature for a symmetric bilinear map on Euclidean or Hilbert space, the conditions when this set contains strongly umbilical, conformal nullity, etc. linear subspaces. The main goals are estimates from above of the codimension of these subspaces for a symmetric bilinear map with positive normal curvature and the inequality type restriction on the extrinsic qth scalar curvature. We estimate from above the codimension of asymptotic and relative nullity subspaces for a symmetric bilinear map with nonpositive extrinsic qth scalar curvature.

The aim of the paper is to extend the notion of $\alpha$-geometry in the classical and in the noncommutative case by introducing a more general class of pull-back metrics and to give concrete formulas for the scalar curvature of these Riemannian manifolds. We introduce a more general class of pull-back metrics of the noncommutative state spaces, we pull back the Euclidean Riemannian metric of the space of self-adjoint matrices with functions which have an analytic extension to a neighborhood of the interval $]0,1[$ and whose derivative are nowhere zero. We compute the scalar curvature in this setting, and as a corollary we have the scalar curvature of the classical probability space when it is endowed with such a general pull-back metric. In the noncommutative setting we consider real and complex state spaces too. We give a simplification of Gibilisco's and Isola's conjecture for the first nontrivial classical probability space and we present the result of a numerical computation which indicate that the conjecture may be true for the space of real and complex qubits., Comment: 18 pages