Following Donaldson's openness theorem on deforming a conical Kähler-Einstein metric, we prove a parabolic Schaudertype estimate with respect to conical metrics. As a corollary, we show that the conical Kähler-Ricci flow exists for short time. The key is to establish the relevant heat kernel estimates, where we use the Weber formula on Bessel function of the second kind and Carslaw's heat kernel representation in [8].

We prove that the conical Kähler-Ricci flows introduced in [17] exist for all time t ∈ [0, +∞). These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class C 1,β is negative or zero, the corresponding conical Kähler-Ricci flows converge to Kähler-Einstein metrics with conical singularities exponentially fast. To establish these results, one of our key steps is to prove a Liouville type theorem for Kähler-Ricci flat metrics (which are defined over C n ) with conical singularities.

In this note we prove that any four-dimensional half conformally flat gradient steady Ricci soliton must be either Bryant's soliton or Ricci flat. We also classify four-dimensional half conformally flat gradient shrinking Ricci solitons with bounded curvature.

It is widely recognized that constructing multiple interface structures for enhanced interface polarization is beneficial to microwave absorption. Here, we report our work of achieving excellent microwave-absorption performance and controlling better-defined interfaces in vertically stacked two-dimensional (2D) MoS with other dimensional building blocks. The optimal reflection loss and effective absorbing bandwidth (reflection loss <-10 dB) of several mixed-dimensional van der Waals heterostructures are as follows: (i) for 2-0 type (2D MoS/zero-dimensional Ni nanoparticles), -19.7 dB and 2.92 GHz; (ii) for 2-1 type (2D MoS/one-dimensional carbon nanotubes), -47.9 dB and 5.60 GHz; and (iii) for 2-3 type (2D MoS/three-dimensional carbon layers), -69.2 dB and 4.88 GHz. As a result, by selected synthesis of different types of microstructures, we can regulate and control microwave-absorption properties in MoS mixed-dimensional van der Waals heterostructures. In addition, attributing to the better-defined interfaces generated in mixed-dimensional van der Waals heterostructures, we found an alternative strategy to improve microwave attenuation properties of 2-0, 2-1, and 2-3 samples by controlling interfacial contacts. The results indicate that mixed-dimensional van der Waals heterostructures provide a new stage for the next generation of microwave-absorbing materials.