Omori–Yau maximum principles, $$V$$ V -harmonic maps and their geometric applications

Parabolic Omori–Yau Maximum Principle for Mean Curvature Flow and Some Applications

“…Remark 3. Note that the above theorem is stronger than Theorem 5 in [2], where they assume that f is bounded above (which corresponds to our case when α = 0). We remark that the condition can be weaken to lim x→∞ u(x) | F (x)| + 1 = 0, since the Laplacian of the function | F | 2 satisfies better estimates: ∆| F | 2 ≤ 2n.…”

“…In this section, we improve Theorem 5 in [2] using Theorem 1.2. The proof is more intuitive in the sense that we use essentially the fact that a self-shrinker is a self-similar solution to the mean curvature flow (possibly after reparametrization on each time slice).…”

“…[3], [10]. On the other hand, various Omori-Yau type maximum principles have been proved for other elliptic operators and on solitons in geometric flows, such as Ricci solition [2] and self-shrinkers in mean curvature flows [4]. The Omori-Yau maximum principles are powerful tools in studying noncompact manifolds and have a lot of geometric applications.…”

Rigidity theorems of spacelike entire self-shrinking graphs in the pseudo-Euclidean space

“…Under the global conditions of Lagrangian entire graph or complete with the induced metric, there are plenty of related works, see e.g. [1,8,9,16,26,29]. It should mention that Chen-Qiu [10] proved that only the affine planes are the complete m-dimensional spacelike self-shrinkers in the pseudo-Euclidean space R m+n n .…”

On VT-harmonic maps

“…:= 2 × [0, T ] and denote the parabolic boundary of2 T by∂ p := ( 2 × {0}) ∪ (∂ 2 × [0, T ]).For the heat flow of magnetic harmonic maps∂ t u = τ (u) + Z (du(e 1 ) ∧ du(e 2 )), Let u 1 , u 2 ∈ C 0 ( 2 , N )be two solutions of heat flow Eq. (5.3) for magnetic harmonic maps into a geodesic ball B R ( p).…”