Let $R = \mathbb{C} [ x_1, x_2, \dotsc , x_n ] / (f)$ where $f$ is a weighted homogeneous polynomial defining an isolated singularity at the origin. Then $R$ and $\operatorname{Der} (R,R)$ are graded. It is well-known that $\operatorname{Der} (R,R)$ does not have a negatively graded component. Wahl conjectured that this is still true for $R = \mathbb{C} [ x_1, x_2, \dotsc, x_n] / (f_1, f_2, \dotsc, f_m)$ which defines an isolated, normal and complete intersection singularity and $f_1, f_2, \dotsc, f_m$ weighted homogeneous polynomials with the same weight type $(w_1, w_2, \dotsc, w_n)$. Here we give a positive answer to the Wahl Conjecture and its generalization (without the condition of complete intersection singularity) for $R$ when the degree of $f_i, 1 \leq i \leq m$ are bounded below by a constant $C$ depending only on the weights $w_1, w_2, \dotsc, w_n$. Moreover this bound C is improved when any two of $w_1, w_2, \dotsc, w_n$ are coprime. Since there are counter-examples for the Wahl Conjecture and its generalization when $f_i$ are low degree, our theorem is more or less optimal in the sense that only the lower bound constant can be improved.

We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair $(7, 8)$ is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and Münzner. This completes the classification of isoparametric hypersurfaces in spheres that É. Cartan initiated in the late 1930s.

We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler–Ricci soliton. In particular, it follows that for any $n \in \mathbb{N}_0$ and for any negative line bundle $L$ over a compact Kähler manifold $D$, the total space of the vector bundle $L^{\oplus (n+1)}$ admits a unique AC expanding gradient Kähler–Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if $c_1 \Bigl ( K_D \oplus {(L^\ast)}^{\oplus (n+1)} \Bigr ) \gt 0$. This generalizes the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient Kähler–Ricci solitons on $\mathbb{C}^n$ with positive curvature operator on $(1, 1)$-forms is path-connected.

Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces − minimal isoparametric hypersurfaces and focal submanifolds. By virtue of their rich structures, we find new series of minimizing cones. They are cones over focal submanifolds and cones over suitable products among these two classes. Except in low dimensions, all such cones are shown minimizing.