Michael-Simon type inequalities in hyperbolic space H n + 1 Hⁿ⁺¹ via Brendle-Guan-Li’s flows

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> Hⁿ⁺¹ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> Hⁿ⁺¹ ,” (Preprint)) as follows (0.1) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:msqrt> <m:mrow> <m:msup> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msubsup> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>+</m:mo> <m:mo stretchy="false">|</m:mo> <m:msup> <m:mrow> <m:mi>∇</m:mi> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:msup> <m:mi>f</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msqrt> <m:mo>−</m:mo> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:mfenced close="⟩" open="⟨"> <m:mrow> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>∇</m:mi> </m:mrow> <m:mo>̄</m:mo> </m:mover> </m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mi>f</m:mi> <m:msup> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> </m:mrow> </m:mfenced> <m:mo>,</m:mo> <m:mi>ν</m:mi> </m:mrow> </m:mfenced> <m:mo>+</m:mo> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:mi>f</m:mi> <m:mo>≥</m:mo> <m:msubsup> <m:mrow> <m:mi>ω</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:msubsup> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:msup> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:msup> </m:math> $${M}{∫ }{λ }√{{f}²{E}₁²+ {∇ }Mf{ }²}-{M}{∫ } ∇ ̄(f{λ }),ν +{∂ M}{∫ }f≥ {ω }ₙ1/n{({M}{∫ }{f}n/n-1)}n-1/n$$ provided that M is h -convex and f is a positive smooth function, where λ ′( r ) = cosh r . In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in (“A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold,” Commun. Pure Appl. Math. , vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the k th mean curvatures in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> Hⁿ⁺¹ by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> Hⁿ⁺¹ ,” (Preprint)) as below (0.2) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mtable class="align" columnalign="left"> <m:mtr> <m:mtd columnalign="right"/> <m:mtd columnalign="left"> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:msqrt> <m:mrow> <m:msup> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msubsup> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>+</m:mo> <m:mo stretchy="false">|</m:mo> <m:msup> <m:mrow> <m:mi>∇</m:mi> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:msup> <m:mi>f</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msubsup> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:msqrt> <m:mo>−</m:mo> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:mfenced close="⟩" open="⟨"> <m:mrow> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>∇</m:mi> </m:mrow> <m:mo>̄</m:mo> </m:mover> </m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mi>f</m:mi> <m:msup> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> </m:mrow> </m:mfenced> <m:mo>,</m:mo> <m:mi>ν</m:mi> </m:mrow> </m:mfenced> <m:mo>⋅</m:mo> <m:msub> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:mi>f</m:mi> <m:mo>⋅</m:mo> <m:msub> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"/> <m:mtd columnalign="left"> <m:mspace width="1em"/> <m:mo>≥</m:mo> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo>◦</m:mo> <m:msubsup> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>W</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>k</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:msup> <m:mo>⋅</m:mo> <m:msub> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:msup> </m:mtd> </m:mtr> </m:mtable> </m:math> {align} & {M}{∫ }{λ }√{{f}²{E}ₖ²+ {∇ }Mf{ }²{E}ₖ₋₁²}-{M}{∫ } ∇ ̄(f{λ }),ν · {E}ₖ₋₁+{∂ M}{∫ }f· {E}ₖ₋₁ \\ & ≥ {({p}ₖ{∘}{q}₁⁻¹({W}₁({Ω})))}1/n-k+1{({M}{∫ }{f}n-k+1/n-k· {E}ₖ₋₁)}n-k/n-k+1 {align} provided that M is h -convex and Ω is the domain enclosed by M , p k ( r ) = ω n ( λ ′) k −1 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>W</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:mfrac> <m:mo stretchy="false">|</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">|</m:mo> </m:math> W₁(Ω)=1/n M , λ ′( r ) = cosh r , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>W</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mfenced close=")" open="("> <m:mrow> <m:msubsup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:mfenced> </m:math> q₁(r)=W₁(Sᵣⁿ⁺¹) , the area for a geodesic sphere of radius r , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> q₁⁻¹ is the inverse function of q 1 . In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in (“Locally constrained curvature flows and geometric inequalities in hyperbolic space,” Math. Ann. , vol. 382, nos. 3–4, pp. 1425–1474, 2022).