| Summary: | We prove the upper and lower bounds of the diameter of a compact manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo form="prefix">dim</mo><mi mathvariant="double-struck">R</mi></msub><mi>M</mi><mo>=</mo><mi>n</mi><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>3</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and a family of Riemannian metrics <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfying some geometric flows. Except for Ricci flow, these flows include List–Ricci flow, harmonic-Ricci flow, and Lorentzian mean curvature flow on an ambient Lorentzian manifold with non-negative sectional curvature.
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