Let $n\ge3$ and let $h:\A(r,1)\to\A(R,1)\subset\mathbb R^n$ be an onto homeomorphism with harmonic coordinate functions. We prove the sharp Nitsche bound \[ R\le R_{n,+}(r):=\frac{nr}{n-1+r^n}, \] and, when $h$ interchanges the two ends, the strictly stronger sharp bound \[ R\le R_{n,-}(r):=\frac{nr^{n-1}}{1+(n-1)r^n}. \] Both critical cases are rigid: equality forces, up to an orthogonal transformation, the corresponding end-preserving or end-reversing radial harmonic homeomorphism. No continuous extension to the closed annulus, boundary homeomorphism, boundary Jacobian, or sign condition on the Jacobian is assumed. The proof converts the nonzero degree of each interior direction map into a probability coupling and establishes a sharp contraction principle for vector measures under positive zonal kernels, using the strict concavity of spherical-cap barycenters. At either critical value, a second-order endpoint defect forces equality for a limiting transfer kernel, whose equality classification yields an orthogonal coupling graph. The remaining trace is locked by a Dirichlet-to-Neumann spectral gap in the end-preserving case and by endpoint H\"older regularity and uniform convergence of the direction maps in the end-reversing case.
Let $r_n$ be the infimum of \[ \frac{\lVert P'-P'(0)P\rVert_{H^2}}{\lVert P\rVert_{H^2}} \] over all degree-$n$ polynomials $P$ satisfying $P(0)=1$ whose zeros lie in the closed unit disk. We prove the quantitative residual bound \[ r_n\geq \exp\!\bigl(-(1+o(1))\sqrt n\log n\bigr) \qquad(n\to\infty). \] As an application, we answer Erd\H{o}s Problem 973 on exterior power sums in the negative, in a form quantitatively stronger than the answer first obtained by Luo, Yang, and Zhu.
Xiaojun Tan, Qihang Wang, Wei Huang et al.· 0 citations
Let $(M^{n+1},g)$ be a compact Riemannian manifold with boundary. Under the assumptions $\Ric_g\geq ng$ and $\II_g\geq0$, Wang proposed a sharp strengthening of the Choi--Wang--Reilly estimate, asserting that the first nonzero Laplace eigenvalue of the boundary is at least $n$; see [J. Geom. Anal. 31 (2021)]. We disprove this assertion in every dimension $n+1\geq3$. More precisely, we construct a sequence of metrics on the hemisphere $\Sph^{n+1}_{+}$ converging in $C^\infty$ to the round metric and satisfying \[ \Ric_g>n g,\qquad \II_g>0,\qquad \lambda_1(\partial\Sph^{n+1}_{+},g|_{\partial\Sph^{n+1}_{+}})<n. \] The construction starts from Zhu's infinitesimal conformal deformation, which lowers one branch of the first boundary eigenspace while preserving the normalized Ricci lower bound to first order. We add a multiple of the spherical height function.
We establish the rigidity statement in the equality case of the hyperspherical-radius inequality of Brendle, Tsiamis, and Wang for compact spin fill-ins with scalar curvature bounded below. More precisely, let $(M^{n\geq 3},g)$ be a compact, connected Riemannian spin manifold having a connected boundary $\Sigma$ and scalar curvature satisfying $\mathrm{scal}_g\geq -n(n-1)$. We prove that equality in the upper bound \[ \inf_{\Sigma}H\leq (n-1)\sqrt{1+\operatorname{Rad}(\Sigma)^{-2}} \] given by Brendle, Tsiamis, and Wang holds if and only if $(M,g)$ is isometric to a geodesic ball in hyperbolic space.
Let $V$ be an $n$-dimensional Euclidean vector space, ,where $n\ge 4$, and $\ell = \lfloor\frac{n}{2}\rfloor$. We prove the sharp pointwise estimate \[ q_2(E) \ge -\frac{2(\ell -1)}{3\ell} \mathrm{Scal}(E) \mathrm{Id}_{\Lambda^2V^*} \] for every algebraic curvature tensor $E$ on $V$ with nonnegative sectional curvature. Applying this estimate to the decomposition $\operatorname{Rm}_{g}=K_{\min}I+E$, we obtain the vanishing of $H^2(M; \mathbb{R})$ under a dimension-dependent strict sectional-scalar curvature pinching condition. At the weak endpoint, all harmonic two-forms are parallel. Apart from the flat case, this yields $b_2(M)=0$ in odd dimensions and $b_2(M)\le 1$ in even dimensions. At even-dimensional endpoint, $b_2(M)>0$ forces $(M, g)$ to be isometric, up to scaling, to $\mathbb{CP}^{\ell}$ with its Fubini-Study metric. As a consequence every closed five-dimensional manifold satisfying the strict pinching condition implies is a rational homology sphere. An anisotropic rescaling of the same homogeneous four-frame estimate also gives the sharp pointwise sectional-scalar pinching criterion \[ K_{\min} \ge \frac{n(n-1)}{n^2-n+12}S_0 \quad \Longrightarrow \mathrm{PIC2}. \] The strict pinching places the curvature tensor in the interior of PIC2 and normalized Ricci flow brings it to a positive constant sectional curvature.
Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the even-dimensional case for $n\ge1000$. We complete the remaining even-dimensional cases. Starting from the structural identities in Frankel--Urschel Lemma 2.1, we derive an exact global defect budget and combine binary rounding with Gram projection. A refined ten-row obstruction handles every even $n\ge66$; a finite exact calculation handles $4\le n\le64$, $n\ne6$; and a separate multi-column energy argument treats $n=6$. The order-two case follows from a direct calculation. The new even-dimensional proof has been formalized in Lean 4, with Frankel--Urschel Lemma 2.1 as its sole external mathematical input. Together with Cheng's odd-dimensional theorem, this proves the S-matrix conjecture in every dimension.
In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations $$\sigma_k(D^2u)=0\quad\text{in}~\mathbb{R}^n\setminus\overline{\Omega}$$ in three dimensional regimes. For $2\le k<n$ and smooth strictly star-shaped domain $\Omega\Subset\R^n$, we establish ball rigidity results in all three dimensional regimes. If $2\leq k<\frac{n}{2}$, the solution with boundary value $-1$ and limit zero at infinity is treated under strict $(k-1)$-convexity of $\partial\Omega$. If $k=\frac{n}{2}$, the solution with boundary value $0$ and logarithmic growth at infinity is considered under strict $(k-1)$-convexity of $\partial\Omega$; if $k>\frac{n}{2}$, the solution with boundary $1$ and fundamental power-growth at infinity is considered under strict $(k-1)$-convexity of $\partial\Omega$. In each case, for a positive constant $c$, $|Du|=c>0$ on $\partial\Omega$ forces $\Omega$ to be a Euclidean ball and determines the solution explicitly. The three arguments are organized by the single parameter $q=(n-k)/k$ and the scale function \[ F_q(t)=\frac{t^{1-q}-1}{1-q},\qquad F_1(t)=\log t. \] After normalization, $U=F_q(v)$ and $M_q[v]=vD^2v-qDv\otimes Dv$ satisfies $\sigma_k(M_q[v])=0$. A unified contact-point calculation gives the sharp bound $|Dv|\le b$;a boundary viscosity contact then yields $H_k\ge qbH_{k-1}$. The integral mechanism that closes the argument changes at the critical exponent: Rellich--Pohozaev identities are used for $q>1$, a scale-invariant Wronskian--Newton current for $q=1$, and a renormalized Newton--Jacobi mass for $q<1$. When $q\leq 1,$ we also obtain a global strict k-convex defining function by Minkowski gauge, this help us extend the exterior construction from convex domains to strictly star-shaped domains.
Zhi-Hui Zhang· 0 citations
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