The question "Can one hear the shape of a drum?" is a classic problem in mathematics (posed by Mark Kac in 1966) that has inspired various related inquiries in different fields. This provoking question which boils down to whether the Laplace spectrum of a manifold could uniquely identify its isometry class. John Milnor swiftly countered this idea by presenting a pair of 16-dimensional tori that were isospectral but not isometric. Subsequently, many mathematicians have pondered the geometric properties that can be deduced from the Laplacian spectrum. ...

The classical Minkowski inequality states that, $\forall \Omega\subset \mathbb R^{n+1}$ convex with $C^2$ boundary, \begin{align} \int_{\partial \Omega} H\ge C_n |\partial \Omega|^{\frac{n-1}{n}}, \label{MI} \end{align} where $H$ is the mean curvature, $c_n$ is a dimensional constant. Equality holds if and only if $\Omega$ is a round ball.

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The following hypersurface flow in $\mathbb H^{n+1}$ was introduced by Brendle-Guan-Li: \begin{align} X_t=\Big(\cosh(\rho) \frac{\sigma_{k}}{\sigma_{k+1}}(\kappa)-\frac{u}{c_{n,k}}\Big)\nu, \label{flow} \end{align} where $\nu$ the outer normal, $\kappa$ the principal curvature vector. The flow preserves $k$-th quermassintegral $\mathcal{A}_{k}$ and decreases $k+1)$-th quermassintegral $\mathcal{A}_{k+1}$. All regularity estimates and convergence would follow if one can establish the gradient estimate (or preservation of starshapedness) for solution of the flow. In turn, Alexandrov-Fenchel inequality for quermassintegrals in hyperbolic space can be established for general starshaped $k+1$-domains. ...

Consider nonlinear transmission systems, $$ \hbox{div} (A(\nabla u)\chi_{D^c} + B(\nabla u)\chi_D) = 0,\tag{1} $$ where $u:B_1\subset {\Bbb R}^n \to {\Bbb R}^m$, and both $A$ and $B$ are strongly elliptic, nonlinear operators. It is well-known that nonlinear systems do not have Lipschitz solutions, in general, even if $A = B$ and the dependence on $\nabla u$ is smooth. This remains true even for minimisers of a nonlinear functional, see [2]. It is also known that the boundary regularity fails for nonlinear systems, even if the boundary data is smooth, see e.g., [3]. However, if we assume that $u$ is Lipschitz up to $\partial D$, then the Lipschitz regularity may have some chances of propagating to the other side, in some small neighborhood, depending on the geometry of $\partial D$. This is because the governing system yields a matching condition of the normal derivatives of $u$ on $\partial D$: formally, $$ A_i^\alpha (\nabla u|_{D^c})\nu_\alpha + B_i^\alpha (\nabla u|_D)\nu_\alpha = 0, $$ whenever the outward normal $\nu$ is defined on $\partial D$. This may leave us in a better situation than a Dirichlet boundary problem, since for the latter problem the normal derivatives of the solution does not need to match those of the boundary data. ...

Here are questions on constant mean curvature (CMC) spheres which arose from the study of the isoperimetric problem in three-dimensional, simply connected, non-compact homogeneous manifolds $X$ diffeomorphic to $\mathbb{R}^3$. ...

Free Boundary Problem refers to an a priori unknown interface, along which a possible phase transition or a qualitative change in the given equations occurs. The subject area has developed in the last 50 years, and has found new branches of directions.

Recently several new research directions have arrived, with many new and challenging problems. ...

In [1], R. Miatello suggests the following problems:

1) Construct congruence lattices which are norm$_1$ and norm$_1*$- isospectral in all dimensions (see [2]).

2) Are there families of $p$-isospectral lens spaces for all $p$, with more than two elements? ...