arXiv subjects

Shuping Huang

Publications and source records attributed to Shuping Huang.

3 recordsLinked to original sources

arXiv2026

The Morse Index, Nullity and Jacobi Fields of Constant-Curvature 2-Spheres in Complex Projective Spaces

We determine the Morse index and normal nullity of minimal immersions $S^2\to \mathbb{C}P^N$ with constant Gauss curvature. For integers $n\ge1$ and $k\in\{0,\ldots,n\}$, the member $\phi_{n-2k,n}:S^2 \to \mathbb{C}P^n$ of the Veronese sequence satisfies \[ \operatorname{Ind}(\phi_{n-2k,n})=2k(n-k)(n+1),\qquad \operatorname{Nul}(\phi_{n-2k,n})=2(n-1)(n+3). \] We also obtain the corresponding formulas for its totally geodesic extensions to $\mathbb{C}P^N$, $N\ge n$, and identify the normal Jacobi kernel with infinitesimal deformations obtained by post-composing the rational normal directrix with projective linear embeddings into $\mathbb{C}P^N$. In particular, every normal Jacobi field is integrable through a family of minimal $2$-spheres.

math.DG↗

arXiv2025

Tight bounds towards Zarankiewicz problem in hypergraph

The classical Zarankiewicz problem, which concerns the maximum number of edges in a bipartite graph without a forbidden complete bipartite subgraph, motivates a direct analogue for hypergraphs. Let $K_{s_1,\ldots, s_r}$ be the complete $r$-partite $r$-graph such that the $i$-th part has $s_i$ vertices. We say an $r$-partite $r$-graph $H=H(V_1,\ldots,V_r)$ contains an ordered $K_{s_1,\ldots, s_r}$ if $K_{s_1,\ldots, s_r}$ is a subgraph of $H$ and the set of size $s_i$ vertices is embedded in $V_i$. The Zarankiewicz number for $r$-graph, denoted by $z(m_1, \ldots, m_{r}; s_1,, \ldots,s_{r})$, is the maximum number of edges of the $r$-partite $r$-graph whose $i$-th part has $m_i$ vertices and does not contain an ordered $K_{s_1,\ldots, s_r}$. In this paper, we show that $$z(m_1,m_2, \cdots, m_{r-1},n ; s_1,s_2, \cdots,s_{r-1}, t)=\Theta\left(m_1m_2\cdots m_{r-1} n^{1-1 / s_1s_2\cdots s_{r-1}}\right)$$ for a range of parameters. This extends a result of Conlon [Math. Proc. Camb. Philos. Soc. (2022)].

math.CO↗

arXiv2025

Graded discrepancy of graphs and hypergraphs

This paper studies the following question of Bollob\'as and Scott: Let $G$ be a graph with $n$ vertices and $p\binom{n}{2}$ edges. What is the smallest $c(p, n)$ such that there is an ordering $v_1, \ldots, v_n$ of the vertices in $G$ with $\left|e(\{v_1, \ldots, v_i\})-p\binom{i}{2}\right|\leq c(p, n)$ for all $i\in \{1,\ldots,n\}$ ? We obtain upper and lower bounds for $c(p,n)$ that are both linear in $n$. Furthermore, we generalize the result to $k$-uniform hypergraphs.

math.CO↗