arXiv subjects

Yunjia Kou

Publications and source records attributed to Yunjia Kou.

4 recordsLinked to original sources

arXiv2026

Endpoint Absolute Monotonicity for the Complete Elliptic Integral of the First Kind

Let \begin{equation*} f_p(x)=\frac{\mathcal{K}(\sqrt{x})}{p-\ln\sqrt{1-x}}, \end{equation*} where $\mathcal{K}$ denotes the complete elliptic integral of the first kind, and set $g_p=1/f_p$. Tian and Yang conjectured that, at $p=\ln4$, both $-f_{\ln4}^{\prime\prime\prime}$ and $g_{\ln4}$ are absolutely monotonic on $(0,1)$. Writing \begin{equation*} \frac{2}πf_{\ln4}(x)=\sum_{n=0}^{\infty}a_n(\ln4)x^n,\quad \fracπ{2}g_{\ln4}(x)=\sum_{n=0}^{\infty}b_n(\ln4)x^n, \end{equation*} we prove \begin{equation*} a_n(\ln4)<0\quad(n\geq3),\quad b_n(\ln4)>0\quad(n\geq0), \end{equation*} thereby settling both conjectures. The proof of the first inequality is based on the representation \begin{equation*} a_n(\ln4)=\frac{(-1)^nC}{15^{n+1}}-M_n, \end{equation*} where $C>0$ and $(M_n)$ is a positive moment sequence. Its strict log-convexity, together with estimates for the initial coefficients, determines the sign of every $a_n$. The second inequality follows from a factorization of the quadratic truncation of the normalized Taylor series and a reciprocal-series argument.

math.CA↗

arXiv2026

On the Rigidity of Closed CMC Hypersurfaces in $\mathbb{S}^5(1)$ with Constant Scalar Curvature

Let $M^4\hookrightarrow\mathbb S^5(1)$ be a closed CMC hypersurface with constant scalar curvature and constant third power sum $f_3=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$. We prove that if $M^4$ has exactly two distinct principal curvatures at some point, then it is isoparametric. More precisely, it is a Clifford torus of the form $\mathbb S^1(r)\times\mathbb S^3(\sqrt{1-r^2})$ or $\mathbb S^2(r)\times\mathbb S^2(\sqrt{1-r^2})$, where $0<r<1$. Under the additional Willmore condition, we obtain a complete classification: every closed CMC Willmore hypersurface $M^4\hookrightarrow\mathbb S^5(1)$ with constant scalar curvature is isoparametric. Consequently, it is congruent to a totally umbilic geodesic sphere, the minimal Clifford torus $\mathbb S^2(1/\sqrt2)\times\mathbb S^2(1/\sqrt2)$, the nonminimal Clifford torus $\mathbb S^1(\sqrt3/2)\times\mathbb S^3(1/2)$, or a Cartan minimal hypersurface. The proofs combine trace-free local tensor identities, an algebraic analysis of the possible principal curvature multiplicities, and a weighted differential $3$-form together with a cut-off argument near the set where principal curvatures coalesce. No sign condition on the scalar curvature is imposed.

math.DG↗

arXiv2026

Closed Minimal Hypersurfaces in $\mathbb{S}^5(1)$ with Constant Scalar and Gauss-Kronecker Curvatures

In this paper, we prove that any closed minimal hypersurface $M^4$ of $\mathbb{S}^5(1)$ with constant scalar curvature and constant Gauss-Kronecker curvature must be isoparametric. Specifically, $M^4$ is either an equatorial 4-sphere, a Clifford torus $\mathbb{S}^2\left(\frac{\sqrt{2}}{2}\right)\times \mathbb{S}^2\left(\frac{\sqrt{2}}{2}\right)$ or $\mathbb{S}^1\left(\frac{1}{2}\right)\times \mathbb{S}^3\left(\frac{\sqrt{3}}{2}\right)$, or a Cartan's minimal hypersurface. Consequently, the squared norm of the second fundamental form $S$ can only take the values 0, 4, 12. This result provides strong support for Chern's Conjecture.

math.DG↗

arXiv2026

Existence of Minimal Homotopies for Immersed Planar Curves

We study the existence of area-minimizing homotopies between homotopic curves in the plane. While the classical Plateau problem establishes the existence of least-area surfaces spanning a single Jordan curve, the corresponding existence theory for homotopies between curves is more subtle and is not directly covered by the same framework. Existing results in the plane are mainly based on combinatorial and algebraic methods, such as decomposing curves into self-overlapping subcurves. These methods are highly effective in the planar setting, but they are often tied to special classes of curves and rely strongly on the local structure of the self-intersections, frequently assuming transverse crossings. In contrast, our approach is geometric and variational, and does not depend on the local structure of the self-intersections. In this paper, we develop a variational existence theory for minimum-area homotopies of immersed planar curves. Our approach adapts classical minimal surface methods by lifting an immersed planar curve with self-intersections into higher co-dimension, where it becomes embedded. For such a lifted curve, we apply Douglas's solution of the Plateau problem to obtain an area-minimizing disk. For closed curves of class $C^1$, we prove uniform convergence of the Douglas minimizers and show that the limiting map minimizes area among all $C^1$ spanning maps of the original planar curve. We then extend the construction to closed Lipschitz curves using approximation and Sobolev compactness arguments. Since the limiting minimizing disk lies in the original plane, it directly produces a null homotopy whose swept area is minimal among all admissible homotopies of the original curve. In this way, the construction connects Plateau theory with minimal homotopy area minimization.

math.GT↗