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We derive a concavity inequality for $k$-Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the $k$-Hessian equations with vanishing Dirichlet boundary and obtain a Liouville-type result. Additionally, we provide new and simple proofs of Guan-Ren-Wang's results on global curvature estimates for $k$-curvature equations., Comment: 30 pages, comments welcome

In this paper, we consider the asymptotic $\sigma_k$ Plateau problem in hyperbolic space. We establish $C^2$ estimates for semi-convex complete hypersurfaces satisfying constant $\sigma_k$ curvature with a prescribed asymptotic boundary at the infinity for $2\leq k\leq n-2$ . The result is based on a new crucial concavity inequality derived for hessian equations., Comment: 38 pages, comments welcome

Robotic fish is one of the most promising directions of the new generation of underwater vehicles. Traditional biomimetic fish often mimic fish joints using tandem components like servos, which leads to increased volume, weight and control complexity. In this paper, a new double-joint robotic fish using a composite linkage was designed, where the propulsion mechanism transforms the single-degree-of-freedom rotation of the motor into a double-degree-of-freedom coupled motion, namely caudal peduncle translation and caudal fin rotation. Motion analysis of the propulsion mechanism demonstrates its ability to closely emulate the undulating movement observed in carangiform fish. Experimental results further validate the feasibility of the proposed propulsion mechanism. To improve propulsion efficiency, an analysis is conducted to explore the influence of swing angle amplitude and swing frequency on the swimming speed of the robotic fish. This examination establishes a practical foundation for future research on such robotic fish systems.

In this paper we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in $\mathbb{R}^{n+1}$ with speed $\psi\sigma_k(\lambda)^{\alpha}$, where $\alpha$ is a positive constant, $\sigma_k(\lambda)$ is the $k$-th elementary symmetric polynomial of the principal radii of curvature and $\psi$ is a preassigned positive smooth function defined on $\mathbb{S}^n$. We prove that under some assumptions of $\psi$, the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known $L^p$ Christoffel-Minkowski problem $u^{1-p}( x ) \sigma_k \left( \nabla^2u+uI\right)=c\psi(x)$ for $p>1$., Comment: 16 pages, 1 figure. comments welcome

In this paper, we study the following prescribed Gaussian curvature problem $$K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^2+|\bar{\nabla}\rho|^2}},$$ a generalization of the Alexandrov problem ($\alpha=n+1$) in hyperbolic space, where $\tilde{f}$ is a smooth positive function on $\mathbb{S}^{n}$, $\rho$ is the radial function of the hypersurface, $\phi(\rho)=\sinh\rho$ and $K$ is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when $\alpha\geq n+1$. Our argument provides a parabolic proof in smooth category for the Alexandrov problem in $\mathbb{H}^{n+1}$. We also consider the cases $2<\alpha\leq n+1$ under the evenness assumption of $\tilde{f}$ and prove the existence of solutions to the above equations., Comment: 31 pages, comments welcome

In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^{\frac{\alpha}{\beta}}\sigma_k^{\frac{1}{\beta}}$, where $\sigma_k$ is the $k$-th elementary symmetric polynomial of the principal curvatures, $\alpha$, $\beta$ are positive constants and $r$ is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of $k$, $\alpha$, $\beta$. When $k\geq2$, $0<\beta\leq 1$, $\alpha\geq \beta+k$, we prove that the $k$-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Li-Sheng-Wang's result from uniformly convex to $k$-convex. When $k \geq 2$, $\beta=k$, $\alpha \geq 2k$, we prove that the $k$-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Ling Xiao's result from $k=2$ to $k \geq 2$., Comment: 25 pages, comments welcome