B-sub-manifolds and Their Stability
نویسنده
چکیده
In this paper, we introduce a concept of B-minimal sub-manifolds and discuss the stability of such a sub-manifold in a Riemannian manifold (M, g). Assume B(x) is a smooth function on M . By definition, we call a sub-manifold Σ B-minimal in (M, g) if the product sub-manifold Σ × S1 is a minimal sub-manifold in a warped product Riemannian manifold (M×S1, g+ e2B(x)dt2), so its stability is closely related to the stability of solitons of mean curvature flows as noted earlier by G. Huisken , S. Angenent, and K. Smoczyk. We can show that the ”grim reaper” in the curve-shortening problem is stable in the sense of ”symmetric stable” defined by K. Smoczyk. We also discuss the graphic B-minimal sub-manifold in Rn+k .
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