Let $n,m$ be integers such that $n\geq2$ and $0\leq m\leq n-2$. Let $C_{m+1}$ denote the $(m+1)$-intermediate curvature introduced by Brendle--Hirsch--Johne. We prove that there are constants $\nu(n,m),C(n,m)>0$ such that the following holds. If $(M^n,g)$ is complete and connected and, for $\delta\geq 0$, \[ \mathrm{Ric}\geq-\delta^2, \qquad C_{m+1}\geq 1, \] then \[ \delta R\leq\nu(n,m) \quad\Longrightarrow\quad \mathrm{Vol} B_R(p) \leq C(n,m)R^m \quad \text{for every $p\in M$ and $R>0$.} \] In particular, taking $m=n-2$ and $\delta=0$ gives Gromov's conjectured codimension-two volume growth estimate under $\mathrm{Ric} \geq0$ and $\mathrm{Scal} \geq1$.
Let $n\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^\gamma(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-\gamma\Delta_g+\mathrm{R}_g$ is strictly positive. We prove that if $n=2$ and $\gamma>0$, or if $n\ge3$ and $0<\gamma \leq 4(n-1...
G. Antonelli, Georg Frenck, Bernhard Hanke· 0 citations
For any complete noncompact manifolds $(M^n, g)$ of nonnegative sectional curvature, with $3\le n\le6$, we obtain $\displaystyle \lim_{r\to\infty}r^{2-n}\int_{B_p(r)}\Sc_g\,dV_g\le8\pi\omega_{n-2}(1-\AVR(M^n,g))$ and the corresponding rigidity. The latter bound and its equality classification hold in $(M^n, g)$ with po...
Let $f:M^n\looparrowright\Sph^{n+q}(1)$, $n\ge2$ and $q\ge1$, be a closed, connected, non-totally-geodesic minimal immersion with second fundamental form $h$, and put $S=|h|^2$ and $S_*=\max_M S$. If $p\in f(M)$ has multiplicity $m$ and $f^{-1}(p)=\{x_1,\ldots,x_m\}$, then \[ \Vol(M)\ge \left[m+\varepsilon_n\sum_{j=1}^...
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on...
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
Let $M_1$, $\cdots$, $M_\ell$ be complete manifolds of the same dimension, where $2\le \ell\in\mathbb{N}$. Suppose that each $M_i$ satisfies two side Gaussian bounds. If some of these manifolds are parabolic and there exists a constant $1\le n_i\le 2$ such that for some $x_i\in M_i$ with $1 \le r \le R<\infty$, $$c_i\l...
Let $X:M^n\to\mathbb R^{n+1}$ be a complete, connected, two-sided minimal immersion without boundary, where $n=3,4$. We prove that finite $\delta$-index, finite Morse index, and finite total curvature are equivalent for $\delta>((n-1)/n)^2$, without assumptions on properness, volume growth, or topology. The main estima...
Marcos Ranieri· 0 citations
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