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• W2963145454 abstract "In this paper, we study entire translating solutions <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>u(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a mean curvature flow equation in Minkowski space. We show that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma equals StartSet left-parenthesis x comma u left-parenthesis x right-parenthesis right-parenthesis vertical-bar x element-of double-struck upper R Superscript n Baseline EndSet> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Sigma ={(x, u(x)) | xin mathbb {R}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a strictly spacelike hypersurface, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma> <mml:semantics> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:annotation encoding=application/x-tex>Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> reduces to a strictly convex rank <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> soliton in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript k comma 1> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^{k,1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (after splitting off trivial factors) whose “blowdown” converges to a multiple <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda element-of left-parenthesis 0 comma 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>lambda in (0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a positively homogeneous degree one convex function in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript k> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also show that there is nonuniqueness as the rotationally symmetric solution may be perturbed to a solution by an arbitrary smooth order one perturbation." @default.
• W2963145454 created "2019-07-30" @default.
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• W2963145454 date "2015-12-22" @default.
• W2963145454 modified "2023-09-30" @default.
• W2963145454 title "Entire downward translating solitons to the mean curvature flow in Minkowski space" @default.
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• W2963145454 doi "https://doi.org/10.1090/proc/12969" @default.
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