Related papers: A diffeomorphism with global dominated splitting c…

We prove that any smooth volume-preserving action of a lattice $\Gamma$ in $\textrm{SL}(n,\mathbb{R})$, $n\ge 3$, on a closed $n$-manifold which possesses one element that admits a dominated splitting should be standard. In other words, the…

Let $M$ be a manifold, $N$ a 1-dimensional manifold. Assuming $r \neq \dim(M)+1$, we show that any nontrivial homomorphism $\rho: \text{Diff}^r_c(M)\to \text{Homeo}(N)$ has a standard form: necessarily $M$ is $1$-dimensional, and there are…

Let $\mathcal{F}$ be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold $M$. We show that if the fundamental group of each leaf of $\mathcal{F}$ has polynomial growth of degree $k$ for some…

We give sufficient conditions for a $ C^1_c $-local diffeomorphism between Fr\'{e}chet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fr\'{e}chet spaces. As a consequence, we…

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal…

We present an example of a twist over a minimal Hausdorff \'etale groupoid such that the restriction of the twist to the interior of the isotropy is not topologically trivial; that is, the restricted twist is not induced by a continuous…

Let $M$ be an $m$-dimensional differentiable manifold with a nontrivial circle action ${\mathcal S}= {\lbrace S_t \rbrace}_{t \in\RR}, S_{t+1}=S_t$, preserving a smooth volume $\mu$. For any Liouville number $\a$ we construct a sequence of…

Given a function $f$ on a smooth Riemannian manifold without boundary, we prove that if $p \in M$ is a non-degenerate critical point of $f$, then a neighborhood of $p$ contains a foliation by spheres with mean curvature proportional to $f$.…

A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant).…

We show that on any smooth compact connected manifold of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing…

We show that for any Riemannian foliation with a simply connected and negatively curved leaf space the normal exponential map of a leaf is a diffeomorphism. As an application, if the leaves are furthermore minimal submanifolds, we give a…

We consider diffeomorphisms in the $C^\infty$-closure of the conjugancy class of translations of the 2-torus. By a theorem of Fathi and Herman, a generic diffeomorphism in that space is minimal and uniquely ergodic. We define a new…

Let $f:Y\to X$ be a finite morphism between Fano manifolds $Y$ and $X$ such that the Fano index of $X$ is greater than 1. On the one hand, when both $X$ and $Y$ are fourfolds of Picard number 1, we show that the degree of $f$ is bounded in…

Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As a consequence of this, we show that the…

Any smooth, closed oriented 4-manifold has a surface diagram of arbitrarily high genus g>2 that specifies it up to diffeomorphism. The goal of this paper is to prove the following statement: For any smooth, closed oriented 4-manifold M,…

We show there is a residual set of non-Anosov $C^{\infty}$ Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. If $M$ is a surface and $2\leq r\leq \infty$, then we will show there exists an open and…

Let $\Omega\subset \mathbb{R}^{n}$ be a bounded open set. Given $1\leq m_1,m_2\leq n-2$, we construct a homeomorphism $f :\Omega\to \Omega$ that is H\"older continuous, $f$ is the identity on $\partial \Omega$, the derivative $D f$ has rank…

In [CKM17], Chodosh, Ketover, and Maximo proved finite diffeomorphism theorems for complete embedded minimal hypersurfaces of dimension $\leqslant$ 6 with finite index and bounded volume growth ratio. In this paper, we adapt their method to…

Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in $\RR^n$ can be regular level sets of a harmonic function modulo a $C^\infty$ diffeomorphism, as well as certain generalizations to…

We find conditions which ensure that the topological complexity of a closed manifold $M$ with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on…