Some Eigenvalues Estimate for the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>ϕ</a:mi> </a:math>-Laplace Operator on Slant Submanifolds of Sasakian Space Forms

This paper is aimed at establishing new upper bounds for the first positive eigenvalue of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"> <a:mi>ϕ</a:mi> </a:math> -Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"> <c:mi>ϕ</c:mi> </c:math> -Laplacian operator on closed oriented <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M4"> <e:mi>m</e:mi> </e:math> -dimensional slant submanifolds in a Sasakian space form <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M5"> <g:msup> <g:mrow> <g:mover accent="true"> <g:mi mathvariant="double-struck">M</g:mi> <g:mo stretchy="true">~</g:mo> </g:mover> </g:mrow> <g:mrow> <g:mn>2</g:mn> <g:mi>k</g:mi> <g:mo>+</g:mo> <g:mn>1</g:mn> </g:mrow> </g:msup> <g:mfenced open="(" close=")"> <g:mrow> <g:mi>ε</g:mi> </g:mrow> </g:mfenced> </g:math> is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M6"> <n:mi>ϕ</n:mi> </n:math> -Laplacian on slant submanifold in a sphere <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M7"> <p:msup> <p:mrow> <p:mi mathvariant="double-struck">S</p:mi> </p:mrow> <p:mrow> <p:mn>2</p:mn> <p:mi>n</p:mi> <p:mo>+</p:mo> <p:mn>1</p:mn> </p:mrow> </p:msup> </p:math> with <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M8"> <s:mi>ε</s:mi> <s:mo>=</s:mo> <s:mn>1</s:mn> </s:math> and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M9"> <u:mi>ϕ</u:mi> <u:mo>=</u:mo> <u:mn>2</u:mn> </u:math> .