Let $n\ge3$ and $D_t=\operatorname{diag}(t^{g_1},\ldots,t^{g_n})$ be a positive diagonal dilation family. We study connected embedded Euclidean hypersurfaces whose diagonal images are minimal. The level-set minimality operator splits into coefficients indexed by the pair sums $g_i+g_j$. Under pair-sum nonresonance, minimality at only $\binom n2$ distinct dilation parameters forces all pair coefficients to vanish. A dimension-reduction argument then shows, without any hypothesis on the coordinate components of the normal, that the second fundamental form vanishes identically. This yields an affine characterization. Repeated-weight helicoidal examples in every dimension and a resonant quadratic cone show that curvature cancellation can survive in genuinely nonuniform families. An application gives a finite-output-level rigidity criterion and an explicit representation for weighted-homogeneous production functions with minimal isoquants.
We study symmetric differentials on weighted hypersurfaces. Let $k$ be an algebraically closed field and let $$ X_d\subset \mathbb{P}_k(a_0,\ldots,a_N),\qquad N\ge3, $$ be a weighted hypersurface, with $a_{\min}:=\min_i a_i$. In characteristic zero we prove twisted vanishing in the range $t<m a_{\min}$ for quasi-smooth...
Let $X_\sigma$ be the affine normal toric variety associated with a full-dimensional strongly convex rational polyhedral cone $\sigma$, and let $m$ be a primitive degree with slice $P_m=\sigma\cap[m=1]$. We construct an explicit flat algebraic family whose completion is universal for equivariant deformations in all deg...
We study how spherical $2$-designs arising from finite group orbits can be completed to spherical $4$-designs by adjoining further orbits, allowing weights in the general theory. For an irreducible real orthogonal $G$-module $W$ with $\mathbb D=\operatorname{End}_G(W)\in\{\mathbb R,\mathbb C,\mathbb H\}$, we consider t...
We study complete
n
-dimensional submanifolds immersed in the Gaussian space
$$\mathbb {G}^{n+p},$$
that is, the Euclidean space
$$\mathbb {R}^{n+p}$$
endowed with its standard metric and the Gaussian weight
$$e^{-|x|^2/4}$$
. More precisely, in codimension
$$p\ge 2,$$
we consider submanif...
W. Barboza, H. D. de Lima, Wallace Gomes· Annali di Matematica Pura ed...· 0 citations
Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption an...
Let $n,d\geq1$, and suppose that $E\subset \mathbb{Q}_p^n$ contains a degree-$d$ polynomial image of $\mathbb{Z}_p$ in every leading direction of a nonempty set $V\subset\mathbb{P}^{n-1}(\mathbb{Q}_p)$. We prove the sharp bound $\dim_H E\geq\dim_H V+1$. For joint block-valued polynomial evaluations, the total incidence...
Yinbo Lou· 0 citations
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