For a smooth, complete and normal metric $g = e^{2u}|dx|^2$ with finite total $n$-th order $Q$-curvature on $\mathbb{R}^n$ with dimension $n \geq 2$, we first show that everywhere non-negativity (resp. non-positivity) $n$-th order $Q$-curvature $Q_g^{(n)}$ implies everywhere non-negativity (resp. non-positivity) of the sectional curvature. Based on this fact, we secondly show that, once $Q_g^{(n)}$ is non-negative, then for any compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial\Omega$, the following sharp isoperimetric inequality holds: $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}} \left(1 - \frac{2}{(n-1)!\,|\mathbb{S}^n|} \int_{\mathbb{R}^n} Q_g^{(n)} \, d\mu_g\right) |\Omega|_g.$$ The third claim in this article is that, if the $n$-th order $Q$-curvature, $Q_g^{(n)}$, is non-positive and under the main assumption that Cartan-Hadamard conjecture holds true, then we have the sharp inequality $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}}|\Omega|_g.$$
Given $n,m\in\mathbb{N}$ such that $n\ge2m\ge4$, letting $g$ be a conformally Euclidean metric on $\mathbb{R}^n$, we consider the question of positivity of the lower-order $Q$-curvatures $Q_g^{(2k)}$ for $k\in\left\{1,\dotsc,m-1\right\}$ when $Q_g^{(2m)}$ is assumed to be nonnegative and not identically zero. We assume moreover that the scalar curvature of the metric $g$ is nonnegative near infinity if $n=2m$ or that $Q_g^{(2m)}$ satisfies a slow decay barrier condition near infinity if $n>2m$. Positive results for this question have been obtained by Gursky and Malchiodi for $m=2$ and $k=1$ in the context of closed manifolds with nonnegative scalar curvature and by Li and Xu and Li, Wei, and Xu for $m\ge2$ and $k\in\left\{1,\min(m-1,2)\right\}$ in the context of conformally Euclidean metrics on $\mathbb{R}^n$. These results hold for all $n\ge2m$. Considering the case where $m\ge4$ and $k=3$, we obtain a positive result for this question when $n\in\left\{2m,2m+1,\dotsc,4m-6\right\}$, namely for these dimensions, we obtain that if $Q_g^{(2m)}\ge0$ and $Q_g^{(2m)}\not\equiv0$ in $\mathbb{R}^n$, then $Q_g^{(6)}>0$. On the other hand, in surprising contrast with the results of Gursky and Malchiodi, Li and Xu, and Li, Wei, and Xu, we find that the answer to this question is negative when $k=3$ and $n\ge N_m$ for some $N_m\in\mathbb{R}$. In this case, we are able to construct examples of conformally Euclidean metrics such that $Q_g^{(2m)}$ is positive everywhere, but $Q_g^{(6)}$ is negative at some point. By stereographic projection, our examples extend to metrics conformal to the standard metric on $\mathbb{S}^n$.
We prove a sharp $2$-systolic inequality for four-dimensional products $\mathbb S^2\times P$, where $P\subset\mathbb R^2$ is an arbitrary convex polygon. Let $h=g_{\mathbb S^2}+g_{\mathrm{eu}}$ be the standard product metric. If a Riemannian metric $g$ on $\mathbb S^2\times P$ has scalar curvature $\geq \sigma>0$, nonnegative mean curvature on every codimension one face, and dihedral angles no larger than the corresponding dihedral angles of $h$, then both its homotopy and homology $2$-systoles are at most $ \frac{8\pi}{\sigma}.$ This confirms a conjecture of Gromov.
For integers $n\ge 1$, consider assertions: $\mathbf{(P_n)}$: Let $\Omega \subset S^n$ be a spherical domain enclosed by totally geodesic $S^{n-1}$'s with non-obtuse dihedral angles. Then in $[0,2(n+1)]$, its Neumann spectrum can only take values among $\{0,n,2(n+1)\}$. Moreover, $n$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is the restriction of a linear function in $\mathbb{R}^{n+1}$, while $2(n+2)$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is restriction of a quadratic polynomial in $\mathbb{R}^{n+1}$. $\mathbf{(Q_n)}$: A weak solution $u$ to $\Delta u = f$ with the Neumann boundary condition, with $f$ H\"older continuous, in a conical polyhedral domain $\Omega$ in $\mathbb{R}^n$ with non-obtuse dihedral angles, is in $C^{2,\alpha}_{loc}(\overline{\Omega})$. We prove the implications \[\mathbf{(Q_n)} \Rightarrow \mathbf{(P_n)},\qquad \mathbf{(P_n)}\Rightarrow \mathbf{(Q_{n+1})}.\] Consequently, both assertions hold in all dimensions. These give the optimal Neumann eigenvalue lower bound and $C^2$ elliptic regularity in non-obtuse Riemannian polyhedral domains.
Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codimension-one refinement of the known almost-everywhere partial regularity, and the exponent is sharp. More generally, for a convex viscosity subsolution of $\sigma_k(D^2u)\ge\lambda>0$, we obtain Hausdorff bounds for strata defined by the affine dimension of all supporting contact sets. The proof combines a support-dependent Chou--Wang barrier argument, an estimate for the product of the smallest $k$ semiaxes of a John ellipsoid, and Mooney's convex section-covering theorem. As a direct analytical consequence, the full distributional Hessian is absolutely continuous and $u\in W^{2,1}_{\mathrm{loc}}(\Omega)$, yielding a $k$-Hessian counterpart of the $W^{2,1}$ regularity known for singular Monge--Amp\`ere solutions. In a logically separate structural part, we characterize the distinguished number of flat directions, $n-k+1$, by an asymptotic infimum mean-value formula over affine sections, and explain how this mean-value heuristic leads to the supporting-contact geometry used in the proof.
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}\lambda_j^+(\Omega)\ge \frac{d-2}{2}.$ It follows that $\lambda_1^+(\Omega)\ge \frac{d-2}{2d},$ with equality if and only if $\Omega$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin--Shtrikman inequality $|\Omega|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partial\Omega}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton's theorem identifies $\Omega$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.
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 $0<\gamma \leq 4(n-1)/(n-2)$, the inclusion $R^0(M)\hookrightarrow R^\gamma(M)$ is a homotopy equivalence, thus generalizing, to all dimensions and in the maximal range, the results of Botvinnik--Rosenberg and Li--Mantoulidis. Then, we prove that if $n\ge3$ and $\gamma>4(n-1)/(n-2)$, the space $R^\gamma(M)$ is contractible, and hence nonempty. This solves a homotopy-theoretic strengthening of a conjecture of Gromov (Conjecture 3, Section 6.1.2,"Four Lectures on Scalar Curvature") in the maximal possible coefficient range. Concerning Gromov's conjecture we also treat the equivariant case and the case of manifolds with boundary.
G. Antonelli, Georg Frenck, Bernhard Hanke· 0 citations