We show that, for all nonnegative integers $k, l, m$ and $ n$, the Yamabe invariant of $$ \#(\mathbb{RP}^3)\# \ell(\mathbb{RP}^2 \#m(S^2 \times S^1)\#n(S^2 \tilde \times S^1)$$ is equal to the Yamabe invariant of $\mathbb{RP}^3$, provided $k + \ell \geq 1$. We then complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of $\mathbb{RP}^3$. More precisely, we show that such maniforlds are either $S^3$ or finite connected sums $ \# m (S^2 \times S^1) \# n (S^2 \tilde \times S^1)$, where $S^2 \tilde \times S^1$ is the nonorientable $S^2$-bundle over $S^1$.

A key ingredient is Aubin’s Lemma [3], which says that if the Yamabe constant is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green’s function for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will al- low us to construct a family of nice test functions on the finite coverings and thus prove the desired result.

We extend recent existence results for compact constant mean curvature normal geodesic graphs with boundary in space forms in a unified way to a large class of ambient spaces. That extension is achieved by solving a constant mean curvature existence problem in a general setting presented in two isometrically equivalent versions.

We consider a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations.

We consider a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations.

Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses Kähler identities, the Rumin-de Rham complex and spectral geometry.

We prove that if a connected Lie group action on a complete Riemannian manifold preserves the geodesics (considered as unparameterized curves), then the metric has constant positive sectional curvature, or the group acts by affine transformations.