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## Scholarly Works (10 results)

In this dissertation we mainly study the geometric structure of vacuum static spaces and some related geometric problems.

In particular, we have made progress in solving the classification problem of vacuum static spaces and in proving the Besse conjecture which is about manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3.

We also extend the local scalar curvature rigidity result of Brendle-Marques-Neves on upper hemisphere to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the work of Corvino. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work of Hang-Wang.

As for generic Riemannian manifolds, we find a connection between Brown-York mass and the first Dirichlet Eigenvalue of a Schrödinger type operator. In particular, we prove a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigate compactly conformal deformations which either increase or decrease scalar curvature. We find local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist for closed or a type of non-compact manifolds with positive scalar curvature. These results together give an answer to a question arises naturally in the work of Corvino and Lohkamp.

Magnesium and its alloys are currently in the spotlight of global research because of the need to limit energy consumption and reduce the environmental impact. In particular, their low densities compared to other structural metals make them a very attractive alternative in the automobile and aerospace industries. However, their low strength compared to other structural materials (e.g. Al and steels) has limited their widespread application.

This dissertation presents the results of developing and investigation of a high strength nanostructured magnesium-aluminum alloy and composite. The nanostructured magnesium alloy is prepared by cryomilling and consolidated by spark-plasma-sintering. Focused ion beam is used to prepare micropillars with different diameters ranging from 1.5 to 8 µm and micro-compression test is conducted by nanoindenter in order to evaluate the mechanical properties. The yield strength obtained in the present study is around three times higher than conventional magnesium alloys (120 MPa vs. 370 MPa). The yield strength of the nanostructured magnesium alloy is further improved through hot extrusion, resulting in a yield strength of 550 MPa and an ultimate strength of 580 MPa. The nanostructured magnesium alloy exhibits a strong size-dependence, and a significant improvement in strength is observed when the pillar diameter is reduced to below 3.5 µm. The deformation mechanisms of the compressed pillars were characterized using transmission electron microscopy. The size-induced strengthening is attributed to a less number of dislocation sources along with a higher activity of non-basal deformation mechanisms.

We have also developed a high strength and thermally stable nanostructured magnesium composite by adding diamantane. A yield strength of 500 MPa is achieved, moreover, excellent thermal stability is demonstrated in the magnesium alloy containing diamantanes. The strength and grain size are thermally stable after annealing at 400°C for 100 hours. In contrast, the yield strength of the alloy without diamantanes decreases significantly after annealing due to severe grain growth. These results suggest that diamantanes are pinning the grain boundaries and inhibiting grain growth at elevated temperatures. Finally, molecular dynamics simulations and finite element analysis are used to explore the deformation mechanisms of magnesium with different grain sizes at atomic resolutions and correct tapering effect on micro-compression test, respectively. The results in the dissertation show that nanostructured Mg-Al alloy and Mg-Al-Diamantane composite are promising materials for aerospace and automobile industries.