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We prove that a smooth compact submanifold of codimension $2$ immersed in $\mathbf{R}^{n},n\geq 3,$ bounds at most finitely many topologically distinct, compact, nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman

We define and compute the energy of 1-foliations on riemannian manifolds. We then derive the Euler-Lagrange equations associated with the energy. We also prove that Riemannian flows on manifolds of constant curvature are ritical if and only if they are isometric. Finally we prove that isometric flows on 3-manifolds are critical if and only if either they are transverse to 2-dimensional foliations or they provide K-contact structures.

It is proved that a convex hypersurface in a Riemannian manifold of sectional curvature at least k is an Alexandrov's space of curvature at least k. This theorem provides an optimal lower curvature bound for an older theorem of Buyalo., 3 pages

We consider the question of when an inequality between lengths of corresponding geodesics implies a corresponding inequality between volumes. We prove this in a number of cases for compact manifolds with and without boundary. In particular, we show that for two Riemannian metrics of negative curvature on a compact surface without boundary, an inequality between the marked length spectra implies the same inequality between the areas, with equality implying isometry.

In this paper we show that if the stability index of $M$ is equal to $n+2$, then the average of the function $|A|^2$ is less than or equal to $n-1$. Moreover, if this average is equal to $n-1$, then $M$ must be isometric to a Clifford minimal hypersurface.

Given a foliation $\mathcal F$ of a compact manifold $M$ and a vector field $X$ on $M$, we provide some necessary and sufficient conditions for $X$ to become the mean curvature vector of (the leaves of) $\mathcal F$ with respect to some Riemannian metric on $M$.

This article uses Cartan-Kähler theory to show that a small neighborhood of a point in any surface with a Riemannian metric possesses an isometric Lagrangian immersion into the complex plane (or by the same argument, into any Kähler surface). In fact, such immersions depend on two functions of a single variable. On the other hand, explicit examples are given of Riemannian three-manifolds which admit no local isometric Lagrangian immersions into complex three-space. It is expected that isometric Lagrangian immersions of higher-dimensional Riemannian manifolds will almost always be unique. However, there is a plentiful supply of flat Lagrangian submanifolds of any complex $n$-space; we show that locally these depend on $\frac{1}{2}n(n+1)$ functions of a single variable.

In this paper, based on Bochner formula, mass decay estimates and elliptic Moser iteration, we show the global existence and asymptotic convergence of a subsequence of solutions of Calabi flow on some closed 3-manifolds, and then the existence of extermal metrics of $L^{2}$-norm of scalar curvature functional on a fixed conformal class is claimed. In particular, we may re-solve part of the Yamabe conjecture on closed 3-manifolds.

1. Introduction. In [B2I a formula is given for evaluating the L2-index of a Dirac-type operator D on a certain class of (noncompact) complete Riemannian manifolds. Although in principle computable, especially in the Fredholm case, this formula contains terms reflecting the contribution of the small eigenvalues, which are difficult to evaluate. We show in this paper that the addition of a skew-adjoint potential V, satisfying reasonable assumptions at infinity, has the effect ofeventually overcoming the influence of the small eigenvalues of D. Thus, the L2-index of the "Dirac-Schr/Sdinger operator" D + 2V, for 2 sufficiently large, is given by an "adiabatic limit" ofr/-invariants and is therefore local at infinity. (See Theorem 3.2 below.) This generalizes and at the same time explains index formulae of Callias type. (See [C], [A].) Due in part to the nature of the problem, but mainly because of the limitations ofthe method we employ, the manifolds we are considering are subject to a number ofconstraints at infinity. Some of these conditions have a clear geometric meaning, but others do not. Thus, the class ofmanifolds to which our results apply is not easy to quantify. It seems possible to enlarge it to encompass all complete manifolds of strictly negative sectional curvature and finite volume; Theorem 3.5 below constitutes an important step in this direction. The L2-index theorem we prove in this paper can be used in conjunction with vanishing type arguments, much in the same way as the standard index theorem for Dirac operators and its relative version are employed in [GL], to gain information about the scalar curvature. To illustrate this, we discuss in Section 4 a "version with boundary" ofthe "conservation principle" for the scalar curvature ofperturbations ofthe standard metric on the n-sphere, suggested by Gromov i-G1 and proved in [L]. We wish to thank Maung Min-Oo for making us aware of these references

The study of finiteness for Riemannian manifolds, which has been done originally by J. Cheeger [5] and A. Weinstein [13], is to investigate what bounds on the sizes of geometrical quantities imply finiteness of topological types, —e.g. homotopy types, homeomorphism or diffeomorphism classes-— of manifolds admitting metrics which satisfy the bounds. For a Riemannian manifold M we denote by RM and KM respectively the curvature tensor and the sectional curvature, by Vol (M) the volume, and by diam(M) the diameter.

Consider a relatively compact region Ω of a Riemann surface R. The term Dirichlet integral over Ω, DΩ(·), is used for the variation whose Euler-Lagrange equation is Δu = 0 on Ω and the term energy integral over Ω, , is used for the variation with Euler-Lagrange equation